LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


ssftm  Class 


gale  Bicentennial 

LIGHT 


pale  'Bicentennial  J&ubucationjs 

With  the  approval  of  the  President  and  Fellows 
of  Tale  University,  a  series  of  volumes  has  been 
prepared  by  a  number  of  the  Professors  and  In- 
structors, to  be  issued  in  connection  with  the 
Bicentennial  Anniversary,  as  a  partial  indica- 
tion of  the  character  of  the  studies  in  which  the 
University  teachers  are  engaged. 

This    series   of  volumes    is    respectfully  dedicated  to 

6raDuate0  of  tfce 


L  IGHT 


A   CONSIDERATION    OF   THE    MORE 

FAMILIAR   PHENOMENA 

OF  OPTICS 


BY 

CHARLES   S.    HASTINGS,   PH.D. 

Professor  of  Physics  in  Yale  University 


NEW   YORK:  CHARLES   SCRIBNER'S  SONS 

LONDON:   EDWARD   ARNOLD 

1901 


Copyright,  1901, 
BY  YALE  UNIVERSITY 


Published,  August,  igoi 


UNIVERSITY   PRESS    •    JOHN   WILSON 
AND   SON    •     CAMBRIDGE,    U.S.A. 


PREFACE 

THERE  is  a  very  large  number  of  phenomena  in  the  domain 
of  optics  which  may  be  regarded  as  familiar,  in  the  sense  that 
all  of  them  may  be  observed  without  the  accessories  which  are 
found  only  in  the  collections  of  philosophical  apparatus  charac- 
teristic of  physical  laboratories.  To  describe  and  explain  this 
class  of  phenomena  is  the  aim  of  this  book.  The  extensive  sub- 
jects of  spectroscopy  and  of  polarized  light  are  excluded  by  the 
limitation  imposed ;  the  remaining  topics  are  presented  rather 
with  a  view  to  usefulness  than  with  respect  to  any  arbitrary 
notion  of  relative  importance.  For  the  reason  just  given  the 
phenomena  of  color  sensations  are  dealt  with  in  a  rather  brief 
way,  since  that  admirable  work,  the  "  Students'  Text-Book  of 
Color,"  by  Professor  Eood,  supplies,  far  better  than  a  less  ac- 
complished writer  could  hope  to  do,  all  demands  of  the  ordinary 
reader.  On  the  other  hand,  so  little  has  been  contributed  during 
more  than  half  a  century  to  what  may  be  called  atmospheric 
optics,  that  it  seems  justifiable  to  extend  the  chapter  treating 
this  subject  to  an  unusual  degree. 

Very  great  improvements  in  the  theory  and  construction  of 
the  most  important  optical  instruments  have  been  made  since 
any  popular  work  devoted  to  their  consideration  has  appeared. 
Indeed,  no  branch  of  applied  science  has  experienced  a  more 
remarkable  revolution  in  recent  times  than  that  which  the 
advance  of  theoretical  optics  has  impressed  upon  the  skilled 

94208 


viii  PREFACE 

optician  of  the  present.  The  history  and  meaning  of  this 
change  is  given  in  the  chapters  on  the  telescope  and  on  the 
microscope. 

Those  readers  who  are  fortunate  enough  to  command  the 
effective  aid  of  mathematical  language  will  find  in  the  appen- 
dices many  rigid  and  easy  proofs  of  important  optical  theorems 
which  must,  in  the  text,  necessarily  rest  upon  more  diffuse 
reasoning.  It  is  hoped  that  the  extraordinary  power  of  the 
method  of  analysis  presented  in  Appendix  A  will  recommend 
it  to  the  student  of  physics. 


CONTENTS 

CHAPTER  I 

WAVE  MOTION  -  REFLECTION  —  REFRACTION 

PAGE 

Definitions  —  Water  waves  —  Huyghens's  principle  —  Reflection  from  plane 
mirrors  —  Reflection  from  spherical  mirrors  —  Refraction  at  a  plane 
surface — Refraction  by  a  plate  —  Refraction  by  a  prism — Total 
reflection  —  Refraction  by  a  spherical  surf  ace  —  Lenses  —  Velocity  of  >^ 

light \r 

CHAPTER  II 

OPTICAL  INSTRUMENTS 

• 

Power  of  a  lens  —  Images  formed  by  lenses  —  Camera  obscura  —  Simple 

microscope  —  Galilean  telescope  —  Keppler's  telescope  —  Compound  / 

microscope  —  Terrestrial  telescope  —  Prism  telescope 19 

CHAPTER  III 

PHENOMENA  OF  LIMITED  WAVE-SURFACES  —  INTER- 
FERENCE—WAVELENGTHS OF  LIGHT 

Diffraction  —  Form  of  the  image  of  a  point  —  Interference  —  Measurement 

of  the  length  of  light  waves  —  Effects  of  perforated  screens     ....      31 

CHAPTER  IV 

DISPERSION  —  CHROMATIC  EFFECTS  OF   DIFFERING 
WAVELENGTHS  — COLORS  OF  THIN  PLATES 

Spectrum  —  Prismatic  series  of  colors  —  Colors  and  wavelengths  —  Inter- 
ference figures  —  Colors    by    interference  —  Thin    plates —  Newton's  S- 
rings 41    * 


x  CONTENTS 

CHAPTER  V 

THE  TELESCOPE 

PAGE 

History  of  the  telescope  after  Galileo  —  Telescopic  discoveries  of  the  seven- 
teenth century  —  Aerial  telescope  —  Inevitable  defect  of  single  lens 
refractors  —  Reflecting  telescopes  —  Herschel's  achievements  —  Modern 
reflectors  —  Achromatic  construction  —  Fraunhofer  —  Modern  refract- 
ing telescopes  and  their  makers  —  Equatorial  mountings  —  Accesso- 
ries  —  Limits  of  magnifying  powers  —  Limit  of  defining  power  —  Art 
of  lens  making  —  Imperfections  of  the  achromatic  objective  —  Best 
limiting  conditions  in  construction  of  telescope  objectives  —  Future 

>ilities  of  the  telescope 53 

CHAPTER  VI 
THE  MICROSCOPE 

History  of  the  microscope  to  1825  —  Theory  of  microscopic  vision  —  High- 
est useful  power  —  Numerical  aperture  —  Denning  power  of  the  eye 
—  History  of  the  achromatic  microscope  —  The  reflecting  microscope  — 
Theory  and  importance  of  the  hemispherical  front  —  Ultimate  defining 
power  of  microscopes  —  Immersion  objectives  —  Abbe's  apochromatic 
construction  —  Compensating  oculars  —  Ratios  of  powers  of  objectives 
to  those  of  oculars  —  Modern  microscope  and  accessories 83 

CHAPTER  VII 
OPTICAL  PHENOMENA  OF  THE  ATMOSPHERE 

Color  of  the  sky  —  Aerial  perspective — Atmospheric  refraction  and  dis- 
persion —  Irregular  atmospheric  refractions  —  A  general  optical  prin- 
ciple —  Wollaston's  experiment  —  Mirage  —  Fata  Morgana  —  Scintil- 
lation—  Coronas  —  Rainbows — Supernumerary  bows — Halos  .  .  .  Ill 

CHAPTER   VIII 
THE  EYE  AND  VISION 

Structure  of  the  eye  —  Accommodation  —  Retinal  images  —  Macula  fovea  — 
Macula  lutea  —  Retinal  structure  —  After-images  —  Chromatic  imper- 
fection —  Stellate  astigmatism  —  Entoptic  phenomena  —  Color  sensa- 
tions —  Young- Helmholtz  theory  —  Methods  of  combining  color 
sensations  —  The  three  fundamental  colors  —  Color  diagram  —  Color 
combinations  —  Color  names  —  Natural  colors  of  objects  —  Color-blind- 
ness —  Holmgren's  test  —  Insufficiency  of  the  Young-Helm holtz  theory  155 


CONTENTS  xi 

CHAPTER   IX 

THEORIES  CONCERNING  THE  NATURE   OF  LIGHT 

PAGE 

The  Newtonian  theory  —  Fresnel  —  Establishment  of  the  wave  theory  — 
Faraday's  discoveries  —  Maxwell's  electromagnetic  theory  —  Confirma- 
tory experiments  of  Hertz 179 

APPENDIX   A 

GENERAL  MATHEMATICAL  THEORY  OF  OPTICAL  INSTRU- 
MENTS .  .' : 189 

APPENDIX   B 

NOTE  ON  SCINTILLATION  AND  ECLIPSE  SHADOW-BANDS    .    209 

APPENDIX   C 
FURTHER  CONSIDERATIONS  ON  HALOS       213 


OF  THE 

UNIVERSITY 

OF 


LIGHT 


CHAPTER   I 
WAVE   MOTION  —  REFLECTION  —  REFRACTION 

THE  word  light  is  used  in  two  distinct  senses,  namely,  to 
designate  the  sensation  which  is  characteristic  of  the  organ  of 
vision,  and  also  as  a  name  for  the  usual  cause  of  that  sen- 
sation. This  double  meaning  of  the  word  would  result  in 
little  inconvenience  if  there  were  always  a  definite  relation 
between  the  sensation  and  its  cause;  but  this  is  far  from 
being  true.  For  example,  when  we  speak  of  white  light,  we 
may  mean  a  certain  sensation  which  is  perfectly  definite  and 
familiar  to  all  seeing  persons,  or  we  may  mean  that  form 
of  energy  which  can  give  rise  to  such  a  sensation.  In  this 
second  sense  the  term  is  wholly  indefinite,  since  there  is  an 
infinite  variety  of  forms  of  energy  which  may  give  rise  to  the 
sensation  of  whiteness.  The  difficulty,  which  is  a  serious  one 
in  scientific  language,  may  be  avoided  by  restricting  the 
meaning  of  the  word  to  one  of  its  significations,  preferably  to 
that  of  a  sensation,  after  the  analogy  of  the  use  of  the  word 
sound.  But  such  a  restriction  would  not  be  in  accordance 
with  well-established  usage,  and  would  necessitate  the  fre- 
quent employment  of  awkward  circumlocutions.  Another 
means  of  avoiding  confusion  is  so  to  divide  the  subjects 
treated  that  the  sense  in  which  the  word  is  used  is  unmis- 
takable. This  method  has  the  advantage  of  conciseness  as 
well  as  that  of  being  in  accordance  with  the  usage  of  most 
writers.  We  shall,  then,  divide  this  book  into  parts,  in  the 
first  of  which  light  will  be  treated  as  a  phenomenon  of  wave 
motion  wholly  independent  of  the  sense  organ  which  betrays 

1 


2  LIGHT 

its  existence  to  us ;  in  the  second  part  we  shall  regard  light 
as  a  sensation,  and  consider  more  particularly  the  relation  of 
the  wave  motion  to  those  sensations  to  which  we  attach  defi- 
nite names.  In  the  first  part  the  eye  is  simply  an  optical 
instrument  quite  like  the  photographic  camera ;  in  the  'second 
part  we  must  study  the  construction  and  orifice  of  the  retina, 
and  the  relation  of  the  mental  impressions  to  the  character  of 
its  stimulations. 

We  are  now  able  to  define  perfectly  the  terms  "white 
light,"  "yellow  light,"  etc.  In  the  first  place,  by  white  light 
we  mean  such  waves  as  are  emitted  by  a  solid  body  at  a 
very  high  temperature,  as,  for  example,  the  incandescent 
lime  in  the  lime-light.  Any  other  kinds  of  waves,  even  if 
indistinguishable  from  these  by  the  unassisted  eye,  are  not 
white  light.  Again,  yellow  light,  green  light,  etc.,  are  the 
simplest  waves  which  will  excite  in  a  normal  retina  the  sensa- 
tions yellow,  green,  etc.  In  the  second  place,  on  the  other 
hand,  these  color  names  designate  certain  familiar  sensations 
which,  as  we  shall  see,  may  be  produced  in  an  indefinite 
variety  of  ways. 

It  is  now  a  little  more  than  two  centuries  since  the  Dutch 
philosopher  Huyghens  published  a  paper  in  which  he  ex- 
plained the  familiar  phenomena  of  light  by  waves  in  a 
medium  that  pervades  all  space  and  is  called  the  luminifer- 
ous  ether.  His  reasoning  was  so  convincing,  the  explana- 
tions so  simple,  and  the  experiments  supporting  his  views  so 
apt,  that  except  for  the  labors  of  the  single  philosopher  then 
living,  who  was  greater  than  Huyghens  himself,  they  could 
hardly  have  failed  to  receive  at  an  early  day  the  universal 
acceptance  which  they  now  command.  Nine  years  earlier, 
in  1669,  Newton  had  commenced  his  labors  in  the  field  of 
optics,  by  which,  largely  on  account  of  fame  and  authority 
won  in  the  domain  of  mechanics  and  astronomy,  he  estab- 
lished a  theory  of  light  which  remained  almost  unquestioned 
for  nearly  a  century  and  a  half.  Newton  supposed  light  to 
consist  in  extremely  small  particles  of  matter  projected  from 
shining  bodies  with  enormous  velocities.  We  now  know 


WAVE  MOTION— REFLECTION  — REFRACTION  3 

that  this  hypothesis  was  not  only  less  fruitful  than  that  of 
Huyghens,  but,  even  with  the  comparatively  limited  range  of 
optical  phenomena  known  to  Newton  and  his  contemporaries, 
was  also  less  probable.  It  was  left  to  Fresnel  —  in  a  re- 
markable series  of  papers  of  the  highest  order  of  merit,  extend- 
ing from  1815  to  1826  —  to  establish  the  wave  theory  upon 
a  foundation  which  leaves  no  room  for  doubt.  Tt  is  true  that 
there  are  many  questions  remaining,  both  as  to  the  character 
of  the  medium  by  which  the  waves  are  transmitted,  and  the 
relations  which  the  medium  bears  to  forms  of  matter  more 
familiar  to  our  senses;  but  there  is  no  more  likelihood  of 
physical  science  in  the  future  rejecting  the  essential  features 
of  the  wave  theory  of  light  than  there  is  of  the  rejection  of 
the  doctrine  of  universal  gravitation,  or  any  other  of  the 
established  laws  of  mechanics. 

As  the  aim  of  this  book  is  to  explain  the  more  familiar 
phenomena  of  light  which  do  not  depend  on  the  form  of  light 
waves,  but  only  on  their  lengths  and  velocities,  we  may 
acquire  all  the  knowledge  of  wave  motion  necessary  for  our 
purposes  by  a  study  of  the  changes  in  the  surface  of  a  liquid, 
such  as  water  or  mercury,  when  agitated  by  a  system  or 
by  systems  of  waves. 

If  a  pebble  is  thrown  into  a  still  pool,  it  will  give  rise  to 
a  series  of  circular  waves  having  their  centre  at  the  place 
of  original  disturbance,  and  each  increasing  in  diameter  at 
an  unchanging  rate.  The  velocity  with  which  the  waves 
move  outward  from  the  centre  is  called  the  wave  velocity;  a 
distance  equal  to  that  from  one  crest  to  the  next  is  called 
the  wavelength;  and  the  time  in  which  a  wave  advances  its 
own  length  is  called  the  period  of  the  wave.  These  terms 
are  applicable  to  all  types  of  waves,  whether  those  in  air 
which  produce  the  sensation  of  sound,  or  those  in  the  light 
ether  which  give  rise  to  the  special  sensation  of  the  organ  of 
vision.  Only  one  other  term  is  necessary  to  define  all  the 
properties  of  a  system  of  waves  required  for  our  purposes. 
Half  the  height  of  the  waves,  that  is,  half  the  difference  of 
level  between  the  tops  of  the  waves  and  the  bottoms  of  the 


4  LIGHT 

troughs,  is  manifestly  the  greatest  distance  that  a  particle  of 
water  in  the  surface  departs  from  its  original  position ;  this 
is  called  the  amplitude  of  the  wave.  It  is  obvious  that  the 
amplitude  of  the  circular  waves  under  consideration  continu- 
ally diminishes,  until  at  a  very  great  distance  it  becomes 
insensible. 

If  two  pebbles  are  thrown  into  the  pool  at  the  same  instant, 
each  becomes  the  centre  of  a  system  of  waves  which  moves 
outward  from  its  centre  exactly  as  though  it  alone  existed. 
This  peculiarity  of  independent  existence  is  the  most  char- 
acteristic feature  of  waves,  and  is  absolutely  without  limita- 
tion. Thus,  on  the  surface  of  the  ocean  we  may  have,  and 
in  general  do  have,  a  vast  number  of  distinct  and  indepen- 
dent systems  of  waves;  there  are  the  three  systems  of  tidal 
waves  —  the  semi-monthly,  diurnal,  and  semi-diurnal ;  the 
waves  produced  by  the  local  winds ;  those  which  have  had  a 
similar  origin  at  a  great  distance  and  pursue  a  different  direc- 
tion of  motion ;  waves  reflected  and  generated  by  a  passing 
ship,  and  every  ripple  caused  by  bird  or  fish  as  perfectly  pre- 
served as  on  still  water.  We  must  not  conclude,  however, 
that  after  our  two  pebbles  are  thrown  every  point  of  the  sur- 
face is  disturbed  twice  as  much  as  it  would  be  with  a  single 
centre  of  disturbance,  although  the  whole  effect  would  be 
obviously  twice  as  great.  There  would  be  places  where  the 
crests  of  one  system  of  waves  would  arrive  at  the  same  mo- 
ment as  the  troughs  of  the  other  system,  and  if  the  two  sys- 
tems were  at  that  point  of  equal  amplitudes,  the  crest  of  the 
one  would  just  suffice  to  fill  the  trough  of  the  other,  thus 
leaving  in  a  limited  area  a  flat  surface  without  waves.  In 
other  places,  again,  one  system  would  act  in  conjunction  with 
the  other  and  produce  waves  of  twice  the  height.  This 
second  characteristic  feature  of  wave  motion  is  called  inter- 
ference. It  can  be  readily  observed  in  all  types  of  waves. 
As  a  common  example,  we  may  detect  regions  of  silence 
about  a  sounding  tuning-fork  if  it  is  held  in  close  proximity 
to  the  ear  and  there  rotated  about  its  axis.  It  is  true  that 
here  the  phenomenon  is  somewhat  more  complicated;  but  we 


WAVE  MOTION— REFLECTION— REFRACTION          5 

may  say,  in  effect,  that  the  system  of  waves  from  one  tine  of 
the  fork  interferes  with  that  from  the  other,  and  hence,  as 
regards  the  air,  the  two  tines  bear  much  the  same  relation 
that  the  two  pebbles  do  in  respect  to  the  surface  of  the  pool, 
So,  on  the  other  hand,  if,  about  two  centres  of  emanations 
of  any  kind,  we  find  regions  where  the  effects  are  mutually 
destructive,  interspersed  with  others  where  they  are  additive, 
we  may  be  sure  that  such  emanations  are  waves.  Just  this 
kind  of  evidence  demonstrates  to  us  that  light  is  a  wave 
motion,  as  will  be  shown  further  on  in  our  study. 

If  we  inquire  why  the  point  where  the  pebble  entered  the 
water  should  be  a  centre  of  a  whole  system  of  waves,  instead 
of  a  single  one  only,  we  may  find  the  answer  in  a  considera- 
tion of  this  kind :  The  surface  of  the  water  is  momentarily  de- 
pressed where  the  stone  enters  it,  but  immediately  the  stone 
passes  through,  and  ceases  to  act  upon  the  surface,  which 
quickly  returns  to  its  original  form  by  the  action  of  weight. 
When,  however,  it  reaches  this  original  form,  it  possesses 
considerable  velocity  as  a  result  of  the  continuous  action 
of  weight,  and  this  velocity  carries  the  surface  beyond  its 
primary  condition  of  flatness,  producing  a  momentary  eleva- 
tion, which,  in  turn,  must  fall  and  send  out  another  wave. 
But  this  consideration  logically  forces  us  to  the  conclusion 
that  every  point  of  the  water  surface  which  is  disturbed  must 
become  the  centre  of  its  own  system  of  waves,  that  is,  that 
every  point  of  every  wave  must  be  regarded  as  such  a  centre. 

It  was  the  recognition  of  this  last  fact  and  its  pursuit 
to  its  legitimate  conclusions  which  constituted  the  notable 
discovery  of  Huyghens,  known  in  the  science  of  optics  as 
Huyghens's  principle.  This  principle  is  of  such  importance, 
and  will  be  so  useful  to  us  in  our  further  studies,  that  it  may 
be  profitably  discussed  here  with  some  care. 

Let  (7,  Figure  1,  be  a  centre  of  disturbances,  and  ab  a 
wave  from  it;  what  will  be  the  final  effect  at  any  other 
point,  as  P?  According  to  the  principle  of  Huyghens,  every 
point  of  ab  must  be  regarded  as  a  centre  of  a  wave  which 
will  finally  reach  P.  From  P  draw  lines  to  different  points 


6  LIGHT 

of  ab,  as  Po^  P02,  Po^  etc.,  such  that  each  succeeding  line 
is  just  one-half  wavelength  longer.  It  is  obvious  that  the 
system  from  the  point  o±  will  be  one-half  wavelength  behind 
the  system  from  03  when  they  reach  the  point  P,  the  crests 
of  the  first  system  corresponding  with  the  troughs  of  the 
second ;  and  thus,  according  to  the  law  of  interference,  their 
effects  at  P  will  be  mutually  destructive.  So  also  for  the 
waves  from  02  and  o^  Further,  it  is  obvious  that  the  effect 
of  each  point  of  the  wave  between  o4  and  oz  will  be  completely 
destroyed  by  the  effect  of  a  certain  point  between  os  and  02, 


FIGURE  1. 

and  so  on.  Since,  however,  as  we  approach  o  the  length  of 
the  segments  into  which  the  wave  is  divided  by  the  lines 
from  P  increases,  there  is  an  unbalanced  effect  of  the  cen- 
tral segments,  and  the  resulting  effect  at  P  is  quite  the  same 
as  though  the  wave  moved  on  parallel  to  itself,  and  only  the 
portion  at  o  were  considered.  (It  is  very  important  to  observe 
that  if  the  length  of  the  wave  ab  were  not  large  compared 
with  the  wavelength,  or  if  the  wave  were  interrupted  so  that 
disturbances  from  all  points  of  ab  could  not  reach  P,  the 
reasoning  and  its  conclusions  would  fall  to  the  ground. 

We  see  from  this  consideration  why  it  is  that  in  a  medium 
which   does   not  change   its   character  the  wave   moves  on 


WAVE  MOTION— REFLECTION— REFRACTION  7 

parallel  to  itself,  the  disturbance  at  o  being  propagated  along 
the  line  Co  to  P,  and  that  at  QI  along  the  line  Col  to  P\  etc. 
If  the  waves  are  light  waves,  these  straight  lines  of  propaga- 
tion are  called  rays  of  light. 

After  having  established  this  fertile  principle,  Huyghens 
could  readily  explain  the  phenomenon  of  reflection  as 
follows :  — 


FIGURE  2. 

Let  C  in  Figure  2  be  a  centre  of  wave  motion  and  AB  rep- 
resent a  mirror,  that  is,  a  barrier  to  the  further  progress  of 
the  waves  which  does  not  destroy  their  motion.  Let  us 
consider  the  state  of  things  at  the  moment  when  a  certain 
wave  ab  first  touches  the  mirror  at  a  point  o.  The  wave 
can  no  longer  move  on  in  its  original  direction;  but  as,  in 
accordance  with  Huyghens 's  principle,  the  barrier  does  not 
destroy  the  motion,  o  must  be  regarded  as  a  centre  of 


8  LIGHT 

disturbances  which  are  propagated  in  all  directions  except 
through  the  mirror.  At  a  certain  instant  afterward,  the  dis- 
turbance from  0  will  be  found  at  o'  in  the  circle  whose  centre 
is  at  p.  A  moment  after  the  wave  touches  the  mirror  at  0 
the  point  ol  in  the  wave  will  reach  the  mirror  at  p^  which 
in  turn  becomes  a  new  centre  of  a  circular  wave.  This  dis- 
turbance will  be  found  in  the  circle  o\  at  the  instant  when 
the  wave  from  p  is  in  0',  since  p}  cfl  is  taken  less  than  po'  by 
the  distance  p^.  In  a  similar  manner  we  may  construct 
the  waves  from  any  number  of  points  on  the  mirror.  In  the 
figure,  five  such  waves  are  given,  the  last,  o'4,  corresponding 
with  p±.  Hence  the  reflected  wave  will  be  found  at  the  given 
instant  to  be  a  circular  wave  o'4  0^  0',  having  (7  as  its  centre, 
as  is  quite  apparent  from  the  construction.  This  point  <7', 
which  is  the  centre  of  the  reflected  waves,  is  called  the  image 
of  C.  If  by  any  means  the  wave  is  so  modified  as  really  to 
have  a  second  point  of  divergence,  that  point  is  called  a  real 
image  ;  but  if,  as  in  this  case,  it  only  seems  to  come  from  such 
a  point,  it  is  called  a  virtual  image. 

This  construction  is  perfectly  general,  subject  only  to  the 
condition  that  the  mirror  shall  be  large  compared  to  the 
wavelength ;  and  it  is  important  not  only  because  it  gives  a 
complete  definition  of  an  optical  image,  but  also  because  it 
contains  the  whole  theory  of  plane  mirrors.  For  example,  let 
D  represent  any  other  source  of  waves,  its  image  will  be  D', 
as  much  nearer  the  mirror  than  C'  as  D  is  than  (7,  and  on 
the  same  side  and  at  the  same  distance  from  the  line  CO 
as  is  Z>,  and  so  on  for  any  number  of  points.  It  will  be 
observed  that,  seen  from  the  mirror,  D  is  on  the  right  of  6y,  but 
the  image  D1  is  on  the  left  of  C' ;  hence  the  image  of  a  system 
of  points  by  a  plane  mirror  is  a  repetition  of  the  system, 
exchanging  only  right  for  left.  Such  a  change  as  this  is 
called  a  perversion,  and  the  image  is  said  to  be  a  perverted 
image. 

Suppose,  now,  that  the  points  C7D,  etc.,  are  either  centres 
of  light  waves  because  self-luminous,  or  centres  of  disturb- 
ance because  light  waves  from  other  sources  fall  upon  them, 


WA  VE  MOTION  —  REFLECTION  —  REFRA  CTION 


9 


and  that  A  B  is  an  unbounded  polished  plane  like  a  looking- 
glass.  It  is  evident  that,  to  the  perception  of  an  eye  in  front 
of  the  mirror,  the  half  of  the  universe  behind  the  mirror  is 
annihilated  and  replaced  by  a  perverted  image  of  the  half  in 
front  of  it.  If  the  mirror  is  not  unbounded,  the  problem  pre- 
sented is  hardly  less  simple :  as  before,  we  must  regard  the 
space  behind  the  plane  of  the  mirror  as  occupied  by  the  per- 
verted copy  of  what  is  in  front;  but,  in  addition,  we  must 
regard  the  mirror  as  a  window  by  which  alone  we  can  see 
into  this  space.  From  these  elementary  considerations  it  is 
obvious  that  we  are  familiar  only  with  left-handed  images  of 
ourselves  in  a  mirror,  and  that  the  smallest  mirror  by  which 
one  can  see  the  whole  figure  has  one-half  the  width  and 
length  of  the  observer,  wholly  independent  of  its  distance. 

A  simple  and  instructive  experiment  is  to  look  into  the 
angle  formed  by  two  vertical  strips  of  looking-glass  wheq 
held  at  right  angles  to  each  other. 
In  this  case,  as  shown  in  Figure 
3,  one  mirror  forms  a  perverted 
image  of  the  region  between  the 
two,  and  the  other  a  perverted 
image  of  the  first  image,  so  that 
in  the  region  between  the  two 
dotted  lines  in  the  figure  we  have 
a  non-perverted  image  of  the  re- 
gion between  the  two  mirrors. 
Thus  an  observer  looking  into 
the  angle  of  the  mirrors  sees  a 
right-handed  image  of  his  face, 

in  which  any  lack  of  symmetry  in  the  features  —  in  the 
arrangement  of  the  teeth,  for  example  —  becomes  very  strik- 
ing to  one  familiar  only  with  the  image  as  seen  in  a  single 
mirror,  although  it  is  obviously  the  appearance  always  pre- 
sented to  his  friends.  So,  too,  a  printed  page  held  in  front 
of  the  angle  of  the  two  mirrors  can  be  read  from  left  to 
right  in  the  reflection,  instead  of  from  right  to  left  as  in 
the  reflection  of  a  single  mirror. 


FIGURE  3. 


10 


LIGHT 


If  the  reflector  is  curved,  the  conditions  are  far  more  com- 
plicated, and,  in  general,  the  reflected  waves  are  no  longer 
spherical,  or,  in  other  words,  no  image  is  formed.  If,  how- 
ever, all  portions  of  the  wave  fall  nearly  perpendicularly 
upon  the  reflector,  the  sphericity  of  the  wave  is  nearly  pre- 
served, and  an  image  is  formed.  If,  as  in  Figure  4,  A,  the 
reflector  ab  is  curved  more  than  the  dotted  line,  which  is 
half-way  in  respect  to  curvature  between  the  wave  and  a  flat 
mirror,  a  real  image  of  0  will  be  formed.  On  the  other  hand, 
if  the  curvature  of  the  reflector  is  less  than  that  of  the  dotted 


c 


c 


FIGURE  4. 

line,  as  is  the  case  in  Figure  4,  B  (which  includes  that  of  a 
reflector  curved  in  the  opposite  direction  or  convex  toward 
(7),  we  shall  have  a  virtual  image  of  C  on  the  opposite 
side  of  the  mirror.  As  no  use  of  the  properties  of  curved 
reflectors  other  than  those  here  described  will  be  made  in 
subsequent  pages,  we  may  content  ourselves  with  the  obser- 
vation (which  may  be  easily  proved,  however)  that,  if  the 
mirrors  are  spherical,  and  the  conditions  are  such  that  the 
image  formed  is  real,  this  image  will  not  be  perverted,  but 
inverted;  again,  if  the  image  is  virtual,  it  will  always  be 
erect  and  perverted,  and  also  smaller  than  the  object. 


WAVE  MOTION— REFLECTION— REFRACTION       11 


Huyghens's  explanation  of  refraction,  that  is,  the  change 
in  direction  of  wave  propagation  when  the  waves  pass  from 
one  medium  to  another,  is  as  simple  as  that  of  reflection,  its 
only  assumption  being  that  there  is  a  definite  and  different 
velocity  of  light  for  each  medium.  To  illustrate  his  expla- 
nation, let  AB,  in  Figure  5,  represent  the  boundary  between 
two  media,  and  0  the  centre  of  wave  motion  in  the  first 
medium.  We  will  suppose  that  the  velocity  in  the  second 
medium  is  only  two-thirds  as  great  as  in  the  first.  Then 
if,  as  in  the  case  of  reflection,  we  consider  the  condition  of 
things  when  a  point  of 
the  wave  o  reaches  the  \c' 

boundary,  we  must  as- 
sume that  this  point  be- 
comes the  centre  of  a 
wave  which  is  propagat- 
ed in  the  second  me- 
dium, and  which  at  a 
certain  instant,  say  after 
the  time  which  is  re- 
quired for  the  point  b  of 
the  wave  to  reach  the 
boundary  .  at  pu  will  be 
found  in  the  circle  o' 
such  that  oo'  equals  f  of 
bpv  If  this  construc- 
tion is  extended  to  all 

intermediate  points  of  the  refracting  surface,  we  shall  find 
that  the  envelop  of  all  the  little  waves  will  be  the  curve 
o'p-H  the  disturbances  inside  this  line  being  mutually  de- 
structive, as  proved  by  the  principle  of  Huyghens.  This 
new  wave  is  not,  in  general,  a  spherical  wave;  hence  we 
do  not  have  an  image  of  C  formed;  but  if  we  impose  the 
condition  that  the  extent  of  the  wave  aob  shall  be  so  re- 
stricted that  it  is  not  far  removed  from  a  straight  line, 
the  refracted  wave  may  be  regarded  as  circular,  and  its 
centre  C'  as  the  image  of  0.  Since  the  refracted  waves 


FIGURE  5. 


12  LIGHT 

do  not  really  come  from  (7',  but  only  appear  to  do  so, 
Ce  is  a  virtual  image  of  0.  The  position  of  the  image  is 
fixed  by  the  considerations,  first,  that  it  must  be  on  the 
perpendicular  Co  because  of  the  obvious  symmetry  of  the 
construction;  and,  second,  that  the  distance  Co  must  be 
three-halves  as  great  as  C'o  because  the  curvature  of  the 
wave  pio'  is  just  two-thirds  what  it  would  have  been  without 
the  change  in  velocity. 

If  the  second  medium  has  another  boundary  A'B',  parallel 
to  the  first  and  at  a  distance  D  from  the  first,  the  waves  in 
passing  through  it  will  have  their  curvature  increased  three- 
halves  times,  and  an  image  of  C"  will  be  formed  at  <?", 
which  is  two-thirds  of  the  distance  C'p'  from  p' ';  but  two- 
thirds  of  Cy  are  equal  to  $  (C'o  +  D)  or  to  Co  +  f  D.  This 
last  distance  is  J.Z)  less  than  the  distance  of  the  primary 
centre  C  from  p'.  It  will  be  observed  that  the  diminution  in 
distance  of  the  apparent  origin  of  the  waves  depends  only  on 
Z>,  not  on  the  distance  of  <7from  the  boundary;  hence,  if  C 
were  in  contact  with  AB,  its  virtual  image  would  still  be  ^  D 
.nearer  AB',  as  in  the  case  considered. 

If  we  apply  the  reasoning  to  light  waves,  we  have  the 
interesting  deductions  that,  since  these  do  in  fact  move  two- 
thirds  as  fast  in  common  glass  as  in  air,  any  point  seen 
through  a  plate  of  glass  appears  at  a  distance  one-third  the 
thickness  of  the  glass  less  than  its  real  distance ;  also  that  a 
plate  of  glass  appears  only  two-thirds  its  real  thickness  when 
one  looks  through  it.  As  light  waves  move  three-quarters  as 
fast  in  water  as  in  air,  it  follows,  by  exactly  the  same  reason- 
ing, that  any  vessel  of  water  appears  only  three-fourths  its 
real  depth  when  we  look  vertically  downward  through  its 
surface. 

A  specially  important  case  in  optics  is  that  where  the  plane 
surface  whence  the  light  emerges  is  not  parallel  to  the  first, 
or  incident  surface.  This  is  illustrated  in  Figure  6.  In  this 
case  the  transmission  through  AB  produces  a  virtual  image 
of  C  at  C'  on  the  perpendicular  Cp,  and  the  transmission 
through  A' B'  a  virtual  image  C'  of  C''  on  the  perpendicular 


WAVE  MOTION  — REFLECTION  — REFRACTION       13 


FIGURE  6. 


line  Cy ,  which  is  one-third  the  distance  from  p'  to  p"  nearer 
the  latter  point  than  the  original  centre  of  waves  is  to  p. 
Thus  it  appears  that  the  effect  of 
such  a  system  is  not  only  to  bring 
the  image  nearer,  but  also  to  dis- 
place it  toward  the  thin  edge  of  the 
wedge,  if,  as  in  our  assumption,  the 
waves  travel  more  slowly  in  the  sec- 
ond medium  than  in  the  first.  In 
optics  a  body  which  acts  upon  waves 
of  light  in  this  manner  is  called  a 
prism. 

Familiar  experience  teaches  that 
whenever  light  passes  from  one  me- 
dium to  another  the  transmission  is 
not  complete;  some  of  the  light  is 
always  reflected.  Thus  the  points 

in  the  boundary  between  two  media  become  the  centres  of 
two  sets  of  waves,  one  of  which,  the  reflected  light,  remains 
in  the  original  medium,  while  the  other  is  the  refracted  wave 
which  we  have  just  been  considering.  The  relative  inten- 
sities of  these  two  sets  are  very  variable,  but  the  reflected 
waves  are  always  stronger  with  increased  obliquity,  so  that 
for  very  large  angles  of  in- 
cidence the  reflected  por- 
tion which  remains  in  the 
first  medium  may  be  even 
stronger  than  the  trans - 
mittedx\There  is  a  singu- 
lar case  of  complete  reflec- 
tion when  the  first  medium 
is  that  in  which  the  ve- 
locity of  propagation  is  the 
slower  of  the  two.  To  FIGURE  7. 

make  this  clear,  let  001,  in 

Figure  7,  represent  a  wave  which  is  moving  inside  of  a  block 
of  glass  in  the  direction  of  the  arrow;  then,  according  to  the 


14  LIGHT 

construction  which  has  been  used  heretofore,  o\o"  will  be  the 
reflected  wave  moving  in  the  direction  of  the  dotted  arrow, 
and  oor  will  be  the  refracted  wave  in  the  outer  medium, 
which  we  may  suppose  to  be  air.  Now  in  this  figure,  if  the 
angle  of  incidence  is  increased  by  a  quite  small  amount,  the 
distance  oof  may  become  just  three-halves  of  oo\,  in  which 
case  the  refracted  wave  would  reduce  to  a  length  zero.  For 
all  larger  angles  the  construction  becomes  impossible.  The 
angle  defined  above  is  called  the  critical  angle,  and  for  it 
and  larger  angles  there  is  no  refracted  light,  all  being  re- 
flected. This  phenomenon  is  called  total  reflection,  and  it  is 
readily  observed  in  a  smooth  glass  containing  water  into 
which  some  object  —  a  spoon,  for  example  —  is  thrust.  If 
one  observes  from  a  level  considerably  below  the  surface  of 
the  water,  a  feeble  image  of  the  object  by  reflection  may  be 
seen,  and  beyond  it  the  part  above  the  water  displaced  by 
refraction;  but  if  the  eye  is  raised  so  as  to  approach  the 
level  of  the  free  surface  of  the  water,  the  reflected  image 
becomes  as  brilliant  as  the  object  itself,  and  nothing  can  be 
seen  beyond  the  surface;  in  short,  the  surface  appears  as 
brilliant  and  as  opaque  as  a  mirror  of  polished  silver. 

An  experiment  to  demonstrate  the  refraction  of  water 
waves  is  very  easily  devised,  and  it  affords  an  interesting 
illustration  of  the  principle  involved.  It  depends  upon  the 
fact  that  the  speed  of  a  water  wave  in  shallow  water  in- 
creases with  the  depth.  If  a  piece  of  flat  board,  say  half 
an  inch  thick,  is  fastened  to  the  bottom  of  a  large  pan,  such 
as  is  used  in  baking  meats,  and  then  water  is  poured  in  until 
the  board  is  covered  to  the  depth  of  an  eighth  of  an  inch,  it 
will  be  found  that  waves  produced  by  immersing  and  remov- 
ing a  rod  at  one  end  of  the  pan  will  move  with  a  diminished 
velocity  over  the  board.  If  the  edge  of  the  board  is  oblique 
to  the  direction  of  the  motion  of  the  waves,  the  wave  follows 
a  new  course  over  the  board;  also,  if  the  board  is  wedge- 
shaped,  it  will  be  found  that  the  direction  of  motion  will  be 
permanently  altered  in  passing  the  shallow  portion. 

It  only  remains  to  consider  what  would  be  the  effect  of  a 


WAVE  MOTION— REFLECTION  — REFRACTION       15 

curved  boundary  between  the  two  media.  In  general,  the 
effect  would  be  complex,  and  no  image  of  the  source  would 
be  formed;  but  with  the  same  limitations  as  those  imposed 
upon  the  reflection  of  light  waves  from  a  curved  surface, 
namely,  that  the  wave  should  nowhere  make  a  large  angle 
with  this  surface,  the  refracted  waves  will  remain  nearly 
circular,  and  their  centre  is  then  an  image  of  the  source.  In 
this  case,  also,  we  will  suppose  the  wave  velocity  one  and 
a  half  times  as  great  in  the  first  medium  as  in  the  second. 
Then,  if  (7,  in  Figure  8,  represents  the  source  and  ApB  the 
curved  boundary  between  the  two  media,  after  the  dis- 
turbance from  the  source  has  reached  the  point  p,  it  will 


A', 

V 

tt 


4 { 

FIGURE  8. 

move  in  the  second  medium  at  two-thirds  its  previous  veloc- 
ity, so  that  when  ol  reaches  pl  it  will  have  reached  0',  if  por 
is  made  equal  to  two-thirds  of  o1pl.  It  is  obvious  that  if  the 
distance  po'  is  less  than  the  distance  of  p  from  the  dotted 
straight  line  in  the  figure,  the  wave  will  be  concave  in  the 
direction  of  its  motion,  and  it  will  move  forward  to  a  point 
which  will  be  a  real  image  of  the  source  C.  This  is  the  case 
in  the  figure.  On  the  other  hand,  if  the  first  distance  is 
greater  than  the  second,  the  wave  will  be  convex  in  the  direc- 
tion of  its  progress,  and  will  appear  to  come  from  a  centre  in 
the  first  medium,  which  therefore  forms  the  virtual  image 
of  the  source.  Whether  the  refracted  wave  shall  be  convex 
or  concave  is  determined  solely  by  the  relation  of  the  curva- 
ture of  the  boundary  to  that  of  the  wave  falling  upon  it,  and 


16  LIGHT 

to  the  relative  velocities  in  the  two  media.  We  may 
the  limiting  case,  namely,  if  the  curvature  of  the  boundary 
corresponds  to  that  of  the  wave  which  falls  upon  it,  the 
shape  of  the  wave  is  not  altered  by  refraction,  and  the  virtual 
image  corresponds  with  the  source  itself.  If  the  curvature 
of  the  boundary  is  greater  than  that  of  the  incident  wave, 
the  latter  is  increased,  and  the  virtual  image  lies  between  0 
and  p  ;  finally,  in  all  other  cases  the  curvature  of  the  wave  is 
decreased,  or  even  possibly  reversed,  if  the  curvature  of  the 
boundary  is  turned  the  other  way.  If  the  effect  is  the  former 
of  these  two,  the  image  formed  is  virtual,  but  more  remote 
from  the  boundary  than  is  the  source ;  in  the  second  case  the 
image  is  real,  and  on  the  opposite  side  of  the  boundary. 

If  the  second  medium  is  bounded  by  another  curve,  so 
that  the  waves  again  enter  a  medium  like  the  first  with  accom- 
panying greater  velocity  of  propagation,  the  change  produced 
in  the  curvature  of  the  waves  will  obviously  be  in  the  oppo- 
site direction  from  that  shown  in  Figure  8.  But  if  the  second 
boundary  is  curved  in  a  reversed  direction,  the  effect  will  be 
additive,  and  the  total  change  will  be  the  same  in  kind  but 
greater  in  amount. 

Applying  these  results  from  the  theory  of  wave  motion  to 
light,  we  require  no  modification  in  the  reasoning,  but  must 
bear  in  mind  that  light  waves  spread  out  in  all  directions 
from  the  centre  of  disturbance,  not  merely  upon  a  surface, 
as  in  the  case  of  water  waves ;  the  boundary  between  the  two 
media  must  then  be  a  surface  instead  of  a  line.  Pieces 
of  transparent  substances,  such  as  glass,  quartz  crystal,  rock 
salt,  etc.,  bounded  by  polished  spherical  surfaces,  are  called 
lenses  ;  they  may  be  regarded  as  the  fundamental  elements  of 
all  optical  instruments.  Lenses  are  divided  into  two  classes : 
those  which  are  thicker  at  the  centre  than  at  the  edge,  called 
positive  lenses,  and  those  which  have  edges  thicker  than  their 
centres,  called  negative  lenses.  The  terms  "positive"  and 
"  negative  "  are  given  because  the  first  kind  may  produce  a  real 
image  of  an  object,  while  the  second  type  never  can  do  so. 
The  shapes  of  the  cross-sections  of  various  lenses  of  each 


WA  VE  MOTION  —  REFLECTION  —  REFRA  CTION       17 


are  shown  in  Figure  9,  the  first  group  being  those  of 
positive  lenses. 

In  order  to  determine  the  effect  upon  a  wave  when  passing 
into  a  new  medium,  it  is  necessary  to  know  the  ratio  of  the 
wave  velocity  in  the  first  medium  to  that  in  the  second. 
This   being  given,  we  can  readily  determine  all  the   phe- 
nomena of  refraction,  however  complicated,  by  following  out 
the   simple   construction  given  in  Figure  5.     In  optics  the 
velocity  of  light  waves  in  air  divided  by  their  velocity  in    // 
any  other  substance  is  called  the  index  of  refraction  of  the// 
substance,  a  quantity  nearly  equal  to  four-thirds  for  water 
and  to  three-halves  for  common  glass.     It  is  an  interesting 


FIGURE  9. 


fact  that  this  ratio  played  as  important  a  r61e  in  Newton's 
theory  of  light  as  in  the  wave  theory;  but  it  was  there  in- 
verted, that  is,  it  was  absolutely  necessary  to  assume  that 
light  travels  faster  in  water  than  in  air.  Thus  there  has 
been  a  theoretically  simple  and  conclusive  method  of  estab- 
lishing one  or  the  other  rival  theory  known  to  philosophers 
ever  since  the  second  one  was  proposed.  In  view  of  the 
fact,  however,  that  light  moves  at  the  rate  of  186,000  miles 
in  a  second,  the  difficulties  of  an  experimental  decision  be- 
tween the  two  theories  seemed  insurmountable ;  but,  finally, 
in  1850,  Foucault,  by  means  of  a  rapidly  rotating  mirror, 
demonstrated  that  the  deduction  from  the  wave  theory  of  light 
is  alone  in  accordance  with  experience.  This  observation 
came  too  late  to  be  of  any  real  philosophical  importance,  for 


18  LIGHT 

the  advocates  of  the  wave  theory  had  already  fought  and  won 
their  battle ;  but  the  investigation  stands  high  among  those 
which  will  always  challenge  the  admiration  of  physicists  on 
account  of  the  skill  exhibited  by  the  experimenter,  involving, 
as  it  did,  the  measurement  of  an  interval  of  time  less  than 
one  one-hundred  millionth  of  a  second. 


CHAPTER  II 
OPTICAL  INSTRUMENTS 

IN  the  last  chapter  the  properties  of  a  lens  have  been 
defined  in  such  a  manner  that  its  power  of  forming  images, 
real  or  imaginary,  is  implied  in  the  definition.  It  is  essential 
to  establish  somewhat  more  definite  notions  in  this  respect. 


FIGURE  10. 

Let  0,  Figure  10,  be  a  thin  double  convex  lens.  Its  effect 
upon  a  wave  from  0  is  to  reduce  its  curvature,  or,  if  suffi- 
ciently powerful,  to  change  the  direction  of  its  curvature  and 
form  a  real  image  of  0  at  <7'.  The  amount  by  which  the 
lens  changes  the  curvature  of  the  incident  wave  is  called  the 
power  of  the  lens.  The  unit  power  is  that  which  changes  a 
flat  wave  to  a  concave  wave  of  ten  inches  radius,  or,  what 
amounts  to  exactly  the  same  thing,  which  can  change  a  con- 
vex wave  of  ten  inches  radius  to  a  flat  one.  This  is  of  course 
a  purely  arbitrary  definition,  but  it  has  long  been  in  use,  and 
we  shall  find  it  convenient  to  retain  it. 

Taking  into  account  the  symmetry  of  both  the  wave  and 
the  bounding  surfaces  of  the  lens  with  respect  to  the  line  Co, 
inspection  of  the  figure  will  show  that  the  image,  wherever 
formed,  must  lie  somewhere  on  that  line  extended.  Con- 
sider, now,  another  system  of  waves  from  the  neighboring 


20  LIGHT 

centre  C^  Since  it  meets  the  first  surface  of  the  lens  at  o 
somewhat  obliquely,  it  will  have  its  direction  changed;  but 
the  refracted  wave  will  meet  the  second  surface,  if  the  lens 
is  thin,  at  the  same  angle  with  which  it  left  the  first  surface ; 
hence,  by  a  reversal  of  the  refraction,  its  original  direction 
will  be  restored,  and  the  only  remaining  change  will  be  that 
of  curvature.  Hence  an  image  of  Ci  will  be  formed  at  C\, 
on  the  extended  line  do.  This  consideration  will  hold  for 
any  number  of  points  near  (7,  and  therefore  an  image  of  the 
region  near  C  is  formed  near  C'.  The  figure  also  renders 
evident  the  fact  that  the  image  is  inverted  if  real,  but  if 
virtual,  that  is,  if  the  centres  of  the  transmitted  waves  are 
on  the  same  side  of  the  lens  ,as  the  original  sources,  the  im- 
age is  erect.  Moreover,  it  is  evident  that  the  ratio  of  the  size 
of  the  image  to  that  of  the  object  is  the  same  as  the  ratio  of 
the  distance  from  the  lens  to  the  image  to  the  distance  of  the 
lens  from  the  object.  In  view  of  the  above  definition  of  the 
power  of  a  lens,  we  see  that  a  powerful  lens  produces  a  small 
real  image  of  a  remote  object,  but  it  may  produce  a  large 
virtual  image  of  a  nearer  object. 

This  reasoning  concerning  the  images  formed  by  lenses 
would  require  modifications  for  thick  lenses,  and  for  those 
which  have  other  than  symmetrical  forms ;  but  the  resulting 
differences  may  be  disregarded  in  this  discussion  without 
affecting  the  validity  of  our  theoretical  conclusions.  It  is 
also  true  that  such  differences  are  generally  small  in  practice. 

If  a  piece  of  white  paper  is  placed  at  Q'  perpendicular  to 
o(7',  and  then  surrounded  by  the  blackened  walls  of  a  box, 
so  that  no  light  except  that  from  the  object  at  C  can  fall 
upon  it,  each  point  of  the  paper  will  appear  to  the  eye 
bright  or  dark,  according  to  the  brightness  of  the  corre- 
sponding point  in  the  object;  in  short,  the  surface  of  the 
paper  will  appear  as  a  faithful,  though  inverted  picture  of 
the  object.  An  instrument  so  constructed  is  called  a  camera 
obscura.  Used  formerly  only  as  a  scientific  toy,  or  occa- 
sionally as  an  aid  in  drawing  the  outlines  of  an  object,  it 
has  become,  since  the  invention  of  the  various  methods  of 


OPTICAL   INSTRUMENTS  21 

fixing  an  image  on  a  sensitive  plate,  one  of  the  most  impor- 
tant of  optical  instruments.  Since  it  is  necessary  to  have 
the  surface  upon  which  the  picture  is  formed  at  a  distance 
from  the  lens  depending  upon  the  distance  of  the  object  to 
be  pictured,  the  sides  of  the  photographer's  camera  box  are 
made  extensible,  like  the  flexible  portion  of  a  bellows,  and 
the  adjustment  is  secured  by  aid  of  the  eye  before  the  sensi- 
tive plate  is  introduced. 

Optically  the  eye  itself  is  simply  a  camera  obscura.  It 
is  figured  in  section  on  page  156,  Figure  40,  and  in  the  eighth 
chapter  a  somewhat  extended  description  of  its  structure  and 
action  is  given.  Here  it  is  only  necessary  to  point  out  the 
one  striking  difference,  from  an  optical  standpoint,  between  it 
and  the  photographer's  camera,  namely,  the  means  for  adapt- 
ing the  apparatus  to  varying  distances  of  the  object  observed. 
Instead  of  modifying  the  distance  of  the  screen  from  the 
lens,  the  power  of  the  lens  itself  is  changed  by  changing  its 
thickness  in  the  middle.  But  as  very  many  eyes  have  either 
lost  this  power  in  part,  or  have  never  possessed  it  in  perfec- 
tion, we  shall  assume  hereafter,  when  an  experiment  is 
described  which  is  to  be  performed  with  the  eye,  either  that 
the  eye  is  a  normal  one  or  that  it  is  rendered  equivalent  to  a 
normal  eye  by  the  employment  of  the  proper  spectacle  lens. 

This  brief  discussion  of  the  optical  properties  of  the  eye 
serves  to  define  the  conditions  of  distinct  vision  of  an  object 
or  of  an  image  of  an  object.  If  object  or  image  is  within 
the  range  of  distinct  vision,  that  is,  in  front  of  the  eye  and 
at  any  distance  greater  than  five  inches,  it  can  be  distinctly 
seen  if  sufficiently  large  and  bright.  In  those  cases  where 
a  definite  value  for  the  distance  of  distinct  vision  is  neces- 
sary for  comparisons,  ten  inches  is  arbitrarily  chosen.  For 
example,  if  we  say  that  the  magnifying  power  of  a  micro- 
scope is  one  hundred  times,  we  mean  that  the  object  seen 
through  the  instrument  appears  a  hundred  times  larger  in 
every  dimension  than  it  would  if  held  at  a  distance  of  ten 
inches  in  front  of  the  unassisted  eye. 

If  an  object  is  too  small  to  be  seen  with  sufficient  distinct- 


22  LIGHT 

ness,  it  may  be  made  to  appear  larger  by  bringing  it  nearer  to 
the  eye,  so  that  at  half  the  distance  it  appears  twice  as  large, 
and  so  on  for  other  approximations.  But  if  this  means  of 
increasing  the  apparent  size  of  an  object  is  carried  too  far,  the 
power  of  the  eye  becomes  insufficient  to  change  convex  wave- 
surfaces  originating  at  the  object  to  concave  ones  having 
their  centres  on  the  retina.  Under  such  circumstances  we 
must  add  to  the  power  of  the  eye  lens  by  employing  a  posi- 
tive lens  close  to  the  eye.  A  lens  so  used  is  called  a  simple 
microscope.  If  the  power  of  this  lens  is  considerable,  that  is, 
sufficient  to  convert  waves  from  a  point  one  inch  or  less  in 
front  of  it  into  flat  waves,  it  is  easy  to  show  that  increased 
size  of  the  image  formed  on  the  retina  is  proportional  to  the 
power  of  the  lens.  It  is  this  relation  which  gives  rise  to  the 
term  "power"  to  designate  that  particular  constant  of  a  lens. 

Since  the  eye  is  optically  a  camera  obscura,  it  follows  that 
the  image  depicted  upon  the  retina  is  inverted  with  respect 
to  the  object.  Why  this  inversion  does  not  appear  in  the 
sensation  is  a  question  of  psychology  rather  than  of  optics. 
It  is  sufficient  for  our  ends  to  recognize  that  an  inverted 
image  on  the  retina  corresponds  to  the  sensation  of  an  erect 
object,  and  vice  versa. 

The  preceding  discussion  of  the  optics  of  the  camera  and 
of  the  eye  is  important  on  account  of  the  fact  that  it  practi- 
cally embraces  the  purely  geometrical  theory  of  all  optical 
instruments.  For  example,  the  magic  lantern  and  the  solar 
microscope  are  cameras  in  which  the  object  is  quite  close  to 
the  lens  and  the  image  remote.  The  only  feature  besides 
this,  and  one  which  is  common  to  both,  is  an  arrangement  by 
which  the  object  can  be  very  strongly  illuminated.  Then, 
also,  the  telescope  and  the  compound  microscope  are  each  a 
combination  of  the  camera  and  the  eye,  the  power  of  the 
latter  being  usually  increased  by  a  lens  which  receives  the 
name  of  eyepiece,  or  ocular.  These  we  shall  consider  some- 
what more  at  length  after  a  description  of  the  optical  prin- 
ciples of  the  opera  glass. 

The  Galilean  telescope,  whether  considered  from  its  his- 


OPTICAL  INSTRUMENTS  23 

tory,  its  achievements,  or  its  theory,  is  one  of  the  most 
interesting  of  all  optical  instruments.  Galileo  learned  in 
1609,  while  visiting  Venice,  that  a  wonderful  instrument 
had  been  invented  the  preceding  year  in  Holland,  which 
would  enable  an  observer  to  see  a  distant  object  with  the 
same  distinctness  as  if  it  were  only  at  a  small  fraction  of 
its  real  distance.  It  required  but  little  time  for  the  greatest 
physicist  of  his  age  to  master  the  problem  thus  suggested  to 
his  mind,  and  after  his  return  to  Padua,  where  he  held  the 
position  of  professor  of  mathematics  in  the  famous  university 
of  that  city,  he  set  himself  earnestly  at  work  making  tele- 
scopes. Such  was  his  success  that  in  August  of  the  same 
year  he  sent  to  the  Venetian  Senate  a  more  perfect  instru- 
ment than  they  had  been  able  to  procure  from  Holland ;  and 
in  January  of  the  next  year,  by  means  of  a  telescope  magni- 
fying thirty  times,  he  discovered  the  four  large  satellites  of 
Jupiter.  This  brilliant  discovery  was  followed  by  that  of 
the  mountains  in  the  moon ;  of  the  variable  phases  of  Venus, 
which  established  the  Copernican  theory  of  the  solar  system 
as  incontestible ;  of  the  true  nature  of  the  Milky  Way,  and 
by  many  others  of  less  philosophical  importance.  Though 
Galileo  did  not  change  the  character  of  the  telescope  as  it  was 
known  to  its  discoverer  in  Holland,  he  made  it  much  more 
perfect,  and,  above  all,  made  the  first  and  most  fertile  appli- 
cation of  the  instrument  to  increase  the  bounds  of  human 
knowledge,  so  that  it  is  inevitable  that  his  name  should  be 
indissolubly  connected  with  the  instrument. 

Considering  the  enormous  interest  excited  throughout  in- 
tellectual Europe  by  the  invention  of  the  telescope,  it  seems 
surprising  that  its  early  history  is  so  confused.  Less  than 
two  years  after  it  was  first  heard  of,  a  discovery  —  perhaps 
the  greatest  in  the  domain  of  natural  philosophy  for  a  thou- 
sand years  —  had  been  made  by  its  means.  Notwithstanding 
these  facts,  the  three  contemporary  or  nearly  contemporary 
investigators  assign  the  honor  of  its  invention  to  three  differ- 
ent persons,  and  if  we  should  write  out  the  names  of  all 
those  to  whom  more  modern  writers  have  attributed  the 


24  LIGHT 

invention  the  list  would  be  a  long  one.  The  surprise  will  be 
somewhat  lessened,  however,  if  we  consider  the  task  before 
a  historian  in  the  next  century  who  undertakes  to  justly 
apportion  the  honor  of  the  invention  of  the  telephone  among 
its  numerous  claimants.  The  analogy,  though  suggested  by 
the  evident  fact  that  the  telephone  is  to  hearing  just  what 
the  telescope  is  to  sight,  could  only  be  close  if  the  future 
historian  were  deprived  of  all  but  verbal  descriptions,  and  if 
contemporary  models  and  diagrams  were  wholly  wanting. 
Under  such  conditions  it  is  difficult  to  believe  that  the  his- 
torian would  easily  escape  antedating  the  discovery  of  the 
telephone  proper  on  account  of  descriptions,  generally  im- 
perfect, of  the  acoustic  telephone.  But  this  would  fairly 
represent  the  condition  of  the  material  at  the  command  of  an 
investigator  of  the  present  day  in  a  question  of  science  of  the 
early  part  of  the  seventeenth  century.  No  wonder,  then, 
that  the  invention  has  been  attributed  to  Archimedes,  to 
Roger  Bacon,  to  Porta,  and  to  many  others  who  have  written 
on  optics ;  but  to  find  the  name  of  Satan  in  the  list  is  cer- 
tainly surprising.  Still  we  read  that  a  very  learned  man  of 
the  seventeenth  century,  named  Arias  Montanus,  finds  in  the 
fourth  chapter  of  Matthew,  eighth  verse,  evidence  that  Satan 
possessed,  and  probably  invented,  a  telescope ;  otherwise  how 
could  he  have  "shown  Him  all  the  kingdoms  of  the  world, 
and  the  glory  of  them  "? 1 

It  seems  to  be  well  established  now,  however,  that  Franz 
Lippershey,  or  Lippersheim,  a  spectacle-maker  at  Middle- 
burg,  was  the  real  inventor  of  the  telescope,  and  that  Galileo's 
first  telescope,  avowedly  suggested  by  news  of  the  Hol- 
lander's achievement,  was  an  independent  invention. 

The  Galilean  telescope  consists  of  a  negative  lens  a,  Figure 
11,  close  to  the  eye,  and  a  positive  lens  b  at  a  certain  dis- 
tance from  the  first,  depending  upon  its  power.  For  the 
sake  of  simplicity  we  shall  suppose  that  the  lens  a  just 
neutralizes  the  lens  of  the  eye.  This  supposition  entails  no 

1  The  early  history  of  the  telescope  is  admirably  treated  in  Poggendorf's 
*  Geschichte  der  Physik/  from  which  some  of  the  foregoing  statements  are  taken. 


OPTICAL   INSTRUMENTS  25 

loss  of  generality  in  our  conclusions,  and  it  is  very  nearly  in 
accordance  with  practice  in  the  instrument  as  it  now  sur- 
vives. In  this  case  the  waves  suffer  no  change  of  curvature 
in  passing  into  the  eye,  and  if,  after  passing  it,  the  lens  b  has 


/v 

\< 

-•• 

T"7      |A 

4«' 

M 

t-  y  iv 

—  4tj«" 

s 

* 

/ 

FIGURE  11. 

such  a  position  that  waves  from  a  distant  source  have  their 
centres  of  curvature  on  the  retina,  the  conditions  of  distinct 
vision  are  met.  Thus  the  effect  of  the  instrument  is  merely 
to  increase  the  virtual  size  of  the  eye,  and  with  it,  as  appears 
from  the  theory  of  the  camera,  the  size  of  the  image  on  the 
retina  in  the  same  ratio.  In  one  particular  only  does  this 
apparatus  —  Galilean  telescope  and  eye  —  differ  from  an  en- 
larged eye,  and  in  that  one  difference  lies  the  limitation  of 
this  form  of  telescope.  In  the  eye  the  iris,  which  limits  the 
portion  of  the  lens  upon  which  light  waves  fall,  is  close  to 
the  lens  itself  and  remote  from  the  retina ;  in  this  magnified 
eye,  however,  the  iris  is  relatively  near  the  retina  and  remote 
from  the  lens.  From  this  it  follows  that  the  light  coming  to 
different  points  of  the  image  passes  through  different  por- 
tions of  the  lens  b.  This  is  evident  from  the  diagram,  for 
the  waves  which  pass  through  the  pupil  of  the  eye  and  con- 
verge to  c'  come  from  that  portion  of  the  objective  b  indi- 
cated by  the  unbroken  part  of  the  wave,  while  the  light 
which  goes  to  form  the  point  c^  comes  from  another  portion 
similarly  indicated.  A  point  in  the  object  which  gives  rise 
to  the  waves  that,  after  passing  through  the  objective,  have 
their  centre  at  c'2,  cannot  be  seen  at  all,  since  these  waves 
are  stopped  by  the  iris.  Thus  the  extent  of  that  portion  of 
an  object  which  can  be  seen  with  such  a  telescope  depends 
upon  the  diameter  of  the  objective  and  of  the  pupil  of  the 


26  LIGHT 

eye.  If  the  magnifying  power  is  considerable,  this  extent  is 
very  small  with  moderate-sized  lenses,  and  large  lenses  are 
not  only  expensive  but  they  are  impracticable  for  short  tele- 
scopes, because  the  wave-surfaces  transmitted  by  them  are  no 
longer  spherical.  An  instrument  having  this  defect  is  said 
to  have-  a  small  field.  But  even  this  field  is  not  illuminated 
quite  to  the  edge,  for  it  is  evident  that  points  on  the  retina 
between  c\  and  c'2  receive  light  from  an  area  of  the  objective 
less  than  that  which  yields  light  to  points  nearer  the  axis  of 
the  eye ;  and  the  more  remote  from  c\,  where  a  portion  of  the 
light  comes  from  the  extreme  margin  of  the  objective,  the 
greater  this  disparity.  Hence  the  field  presents  to  the  eye 
an  outer  marginal  circle  of  indistinct  vision.  For  these 
reasons  this  type  of  telescope  is  adapted  only  for  use  where 
very  moderate  magnification,  say  from  two  to  five  or  six 
times,  is  required,  and  where  extent  of  field  may  be  sacrificed 
to  compactness  and  lightness.  In  opera  glasses,  with  a 
magnification  ordinarily  of  three  or  three  and  a  half  times, 
it  serves  well,  and  finds  a  rival  only  in  the  ingenious  prism 
telescope  which  will  be  described  later.  It  is  also  used  with 
the  sextant,  where  lightness  and  shortness  are  both  very 
desirable;  but  in  almost  all  other  cases  it  is  replaced  by  a 
type  proposed  by  the  famous  astronomer  Keppler,  in  1611. 

/Keppler's  telescope,  ordinarily  called  the  astronomical  tele- 
scope, may  be  readily  understood  from  a  consideration  of  the 
camera  obscura,  as  illustrated  in  Figure  10.  Here  the  objec- 
tive forms  an  image  of  distant  objects  on  the  screen.  Now 
imagine  the  eye  just  at  the  centre  of  the  lens;  if  turned 
toward  the  object,  this  would  appear  of  a  certain  magnitude ; 
if  toward  the  screen,  the  inverted  image  there  depicted 
would  appear  of  the  same  magnitude  as  the  object  itself,  as 
follows  from  the  principle  explained  on  page  20.  If  now 
the  eye  is  brought  nearer  to  the  screen,  the  image  will 
appear  larger  than  the  object  in  the  inverse  ratio  of  the  dis- 
tance of  the  eye  and  of  the  lens  from  the  image.  This  illus- 
trates the  general  principle  involved,  although  in  practice 
the  screen  is  omitted  and  the  eye  is  placed  in  the  prolonga- 


OPTICAL  INSTRUMENTS  27 

tion  of  the  axis  of  the  lens,  because  this  permits  a  larger 
portion  of  the  total  light  transmitted  by  the  objective  to 
reach  the  retina. 

It  thus  appears  that  in  the  astronomical  telescope  the  object 
is  seen  inverted,  a  matter  of  no  importance  in  astronomical 
observations,  and  that  the  magnifying  power  is  equal  to  the 
distance  from  the  objective  to  the  image  divided  by  the  dis- 
tance from  the  image  to  the  eye.  If,  in  order  to  secure  greater 
magnification,  the  eye  is  brought  very  near  the  image,  its 
power  must  be  increased  by  a  lens  used  as  a  simple  micro- 
scope, just  as  in  the  case  of  a  real  object  which  is  too  near 
for  distinct  seeing.  Such  a  lens,  or,  preferably,  system  of 
lenses,  is  called  the  eyepiece,  or  ocular.  The  magnification 
then  becomes  the  power  of  the  ocular  divided  by  the  power 
of  the  objective. 

The  figure  makes  evident  that  in  this  variety  of  telescope, 
unlike  the  one  previously  considered,  light  from  every  por- 
tion of  the  objective  goes  to  every  point  of  the  image  on  the 
retina;  hence  the  size  of  the  field  does  not  depend  at  all  upon 
the  size  of  the  objective,  nor  is  there  any  variation  of  bright- 
ness toward  its  margin.  This  is  an  advantage  of  great 
moment  in  powerful  instruments. 

The  compound  microscope  may  be  conveniently  regarded 
as  an  inverting  telescope  adjusted  for  an  object  very  near 
the  objective.  The  geometrical  principles  involved  in  the 


t 

c 

c' 

FIGURE  12. 

construction  are  illustrated  in  Figure  12,  where  b  is  the 
objective  which  forms  an  image  of  the  object  c  at  <?,  which, 
in  turn,  is  observed  by  means  of  an  ocular  a.  The  size  of  the 


28  LIGHT 

image  is  to  that  of  the  object  as  the  distance  Id  is  to  be,  or 
very  nearly  the  power  of  the  objective  multiplied  by  the 
length  of  the  tube.  This  image,  again,  is  magnified  by 
the  ocular  in  the  ratio  of  the  power  of  the  ocular^  hence  the 
whole  magnification  is  equal  to  the  product  of  the  power  of 
the  ocular  by  the  length  of  the  tube  and  by  the  power  of  the 
objective.  It  is  evident  from  the  figure  that  such  an  instru- 
ment shows  an  inverted  image  of  the  object.  The  compound 
microscope  is  in  no  respect  theoretically  superior  to  the 
simple  microscope,  but  it  is  impracticable  to  make  simple 
microscopes  of  very  great  power,  say  with  a  magnification  of 
much  more  than  250,  because  of  the  extreme  minuteness  of 
the  requisite  lenses. 

If  the  ocular  of  the  astronomical  telescope  is  replaced  by  a 
compound  microscope,  we  shall  see  an  inverted  image  of  the 
image  formed  by  the  objective,  that  is,  an  erect  image  of 
the  distant  object.  This  constitutes  the  terrestrial  telescope, 
or  spyglass.  It  is  necessarily  longer  than  the  astronomical 
telescope  and  less  perfect,  because  it  is  subject  to  the  un- 
avoidable defects  of  a  larger  number  of  lenses  with  their 
accompanying  loss  of  light.  It  is  employed  with  good  reason 
for  the  ordinary  purposes  of  a  spyglass,  but  its  customary  use 
in  surveyors'  instruments  is  to  be  deprecated. 

The  prism  telescope,  which  was  invented  nearly  a  half 
century  ago  by  Porro,  an  Italian  engineer,  has  recently  been 
perfected  mechanically,  and  is  rapidly  replacing  the  simpler 
Galilean  construction  for  powers  from  four  to  twelve.  Its 
principle  and  construction  may  be  understood  from  a  study 
of  Figure  13,  in  which  b  represents  an  objective,  and  ii  an 
inverted  image  of  the  distant  object  to  be  observed.  If  a 
rectangular  prism  p  is  placed  so  that  one-half  of  it  intercepts 
the  light  from  6,  an  image  i\  would  be  formed  by  total  reflec- 
tion from  the  shorter  face  of  the  prism  at  the  place  indicated 
in  the  figure  were  the  light  not  again  reflected  from  the 
second  surface  so  that  a  real  image  is  formed  at  i^  This 
last  image  is  of  course  an  unperverted  repetition  of  ^,  because 
formed  by  a  rectangular  mirror,  but  if  examined  from  the 


OPTICAL  INSTRUMENTS  29 

side  a,  as  might  be  accomplished  either  by  receiving  the 
light  upon  a  white  screen  or  by  reversing  the  direction  of 
the  light  by  a  mirror,  it  would  appear  to  be  perverted  in  the 


k 


FIGURE  13. 

plane  of  the  diagram.  If  now  a  second  prism  _p',  quite  like 
the  first,  but  with  its  base  at  right  angles  to  the  plane  of  the 
paper  instead  of  parallel  with  it,  receives  the  light  from  p  on 
the  lower  part  of  its  largest  surface,  it  will  form,  after  two 
total  reflections,  an  image  z'4  above  the  plane  of  the  figure. 
Thus  by  the  four  successive  reflections  the  final  image  is  a 
complete  inversion  of  i^  and  the  light  is  also  restored  to  its 
original  direction,  whence  the  image  can  be  observed  by  an 
ocular  with  the  eye  turned  toward  the  object.  The  advan- 
tages of  this  construction  are  sufficiently  obvious.  First, 
the  objective  and  ocular  are  essentially  those  of  the  astro- 
nomical telescope,  and  it  retains,  therefore,  the  moderate 
ratio  of  aperture  to  power  and  the  relatively  large  field. 
Second,  the  instrument  is  very  much  shortened,  so  that  it 
is  rendered  conveniently  portable.  On  the  other  hand,  it 
has  defects  which  are  far  from  immaterial.  The  loss  of  light 
from  the  reflections  from  the  larger  faces  of  the  two  prisms 
and  by  absorption  in  their  material  is  by  no  means  incon- 
siderable, and,  as  might  be  supposed,  objects  appear  notably 
less  bright  than  through  a  Galilean  telescope.  Then  the 
awkward  form  necessitated  by  the  offset  in  the  course  of  the 
light  introduces  structural  difficulties  which  doubtless  explain 


30  LIGHT 

the  very  long  time  during  which   the   invention  remained 
undeveloped. 

This  completes  our  review  of  the  purely  geometrical 
principles  involved  in  all  the  more  important  optical  instru- 
ments. It  will  be  observed  that  the  magnification  in  each 
case  depends  very  simply  upon  the  powers  of  the  lenses 
which  enter  into  the  several  constructions;  but  when  we 
come  to  discuss  the  efficiency  of  such  instruments,  we  shall 
find  that  these  powers  play  a  wholly  insignificant  role;  in 
short,  that  the  effective  diameters  alone  are  the  measures  of 
optical  efficiency. 


CHAPTER  III 

PHENOMENA  OF  LIMITED  WAVE-SURFACES  —  INTER- 
FERENCE—WAVELENGTHS OF  LIGHT 

HERETOFORE  we  have  considered  only  those  phenomena 
•which  necessarily  follow  from  the  rectilinear  propagation  of 
waves,  itself  a  consequence  of  Huyghens's  principle,  and 
those  dependent  on  the  varying  velocities  in  different 
media.  We  now  enter  upon  a  consideration  of  the  con- 
sequences entailed  by  limiting  the  extent  of  the  wave-front ; 
and  we  shall  see  that  these  consequences,  very  interesting 
in  themselves,  as  well  as  of  extensive  application  to  the 
explanation  of  phenomena  of  every-day  observation,  are  con- 
nected by  simple  laws  with  the  absolute  value  of  the  lengths 
of  the  waves. 

Suppose  Figure  14  to  represent  a  wave-front  limited  by 
the  screen  ab,  moving  forward  toward  the  centre  p.  The 
point  p  is  thus,  according  to  definition,  the  image  of  the 
point  from  which  the  wave  took  its  origin.  According  to 
the  principle  of  Huyghens,  each  point  of  the  wave-front  ab 
must  be  regarded  as  a  centre  of  disturbances  which  are  propa- 
gated in  all  directions  through  the  medium.  .Consider  the 
condition  at  pi,  so  chosen  that  its  distance  from  a  is  one-half 
a  wavelength  greater  than  its  distance  from  5,  that  is,  so 
that  aaf  equals  a  half  wavelength.  Here  a  disturbance  set- 
ting out  from  a  will  reach  p1  at  the  same  instant  as  one 
starting  from  b  one-half  a  period  later ,  in  short,  the  motion 
derived  from  a  will  be  exactly  equal  and  opposite  to  that 
from  b.  Thus  the  effects  of  these  two  pairs  of  points  in  the 
wave  ab  will  perfectly  neutralize  each  other.  But  these  two 


32  LIGHT 

points  are  the  only  ones  in  the  limited  wave-front  ab  so 
related  to p^\  hence  the  effect  of  other  pairs  of  points,  which 
we  might  imagine  chosen  in  the  wave-front  symmetrically 
placed  with  respect  to  its  middle  point,  will  only  partially 


P 
+, 
•ft 


FIGURE  14. 


counteract  each  other.  For  points  between  p  and  pl  the 
mutual  destruction  of  the  elementary  waves  is  even  less 
complete,  as  there  is  no  pair  of  points  which  wholly  destroys 
each  other's  effect.  Take  now  the  point  p^  such  that  the 
distance  from  a  is  a  whole  wavelength  greater  than  its  dis- 
tance from  b.  In  this  case  we  see  that  the  disturbance  from 
the  middle  point  of  the  wave -front  will  be  a  half  wave- 
length behind  that  from  b,  and  will  thus  destroy  its  effect; 
but  for  every  point  between  b  and  the  middle  of  the  wave- 
front  a  corresponding  point  between  the  middle  and  a  can 
be  found  which  is  a  half  wavelength  further  from  p2;  hence 
the  effects  of  all  the  elementary  waves  at  p2  would  be  nil, 
and  there  the  medium  would  be  perfectly  quiescent.  If  we 
consider  points  more  remote  from  p,  we  shall  find,  by  the 
extension  of  the  same  reasoning,  that  in  general  there  will 
be  motion  due  to  the  effect  of  the  bounded  wave-front, 
except  where  the  difference  of  the  distance  from  a  and  b  is 
a  whole  number  of  wavelengths.  When  this  difference 
is  an  odd  number  of  half  wavelengths,  the  disturbance  is 
greater  than  at  any  closely  neighboring  point,  because  the 
conditions  of  mutual  destruction  are  most  widely  departed 
from;  on  the  other  hand,  it  is  obvious  that  the  absolute 


DIFFRACTION  33 

value  of  the  disturbance  decreases  very  rapidly  in  leaving 
the  position  of  the  geometrical  image  p,  because  a  larger  and 
larger  number  of  pairs  of  mutually  destructive  centres  can 
be  found.  The  light  which  is  found  outside  the  geometrical 
image  is  called  diffracted  light,  and  a  large  class  of  analo- 
gous phenomena  are  embraced  under  the  general  term  of 
diffraction. 

The  conclusion  from  this  study  is,  that  a  limited  concave 
wave-front  forms,  not  a  simple  image  at  its  geometrical 
centre,  as  we  have  assumed  heretofore,  but  a  series  of  images 
of  which  the  middle  one  corresponds  in  place  with  the  geo- 
metrical image,  and  is  by  far  the  strongest,  while  it  is  sym- 
metrically flanked  on  either  side  by  secondary  images  rapidly 
diminishing  in  intensity.  But  we  are  able  to  infer  far  more 
than  this.  It  is  evident  from  the  figure  that  the  distance 
pp%,  which  is  half  the  diameter  of  the  central  image,  bears 
the  same  ratio  to  the  distance  ap  as  aa",  or  one  wavelength, 
does  to  the  distance  ab. 

If  the  waves  are  those  of  light  and  the  wave-front  is  a 
spherical  surface  bounded  by  a  circular  aperture  of  which 
ab  may  be  taken  to  represent  a  diameter,  we  have  the  most 
common  and  most  interesting  case  of  optics,  for  it  is  practi- 
cally that  of  all  optical  instruments.  In  this  case,  although 
the  phenomenon  is  more  complex  in  its  numerical  relations, 
it  does  not  differ  in  kind.  The  image  proper  becomes  a 
circular  area  instead  of  a  point,  and  the  secondary  images 
are  concentric  circles,  as  we  should  expect ;  but  the  distances 
from  the  centre  to  the  various  rings  are  slightly  greater  than 
the  values  derived  from  the  simple  treatment  of  the  last 
paragraph.  For  example,  the  distance  from  the  centre  of 
the  primary  image  to  the  first  ring  is  1.2  times  as  great  as 
from  p  to  pz  in  the  figure.  It  is  this  relation  that  we  shall 
find  of  importance  when  we  come  to  the  consideration  of  the 
limitations  of  optical  instruments. 

Since  the  foregoing  considerations  are  perfectly  general 
and  applicable  to  all  kinds  of  waves,  the  question  naturally 
arises,  Why  do  we  so  rarely  notice  such  extraordinary 

3 


34  LIGHT 

phenomena  in  the  numerous  examples  of  wave  motions  about 
us  ?  The  answer  to  this  question  will  be  .discovered  in  analyz- 
ing the  somewhat  rigid  conditions  implied  by  the  reasoning 
connected  with  Figure  14.  First  of  all,  in  order  to  secure 
regular  phenomena  about  the  region p,  we  must  have  a  series  of 
waves  of  uniform  length,  since  the  distance  from  the  primary 
to  the  secondary  image  depends  upon  this  length.  But 
upon  the  surface  of  water  the  waves  are  generally  of  the 
utmost  irregularity,  nor  do  they  ordinarily  converge  toward 
a  centre,  so  it  is  clear  that  highly  artificial  and  unusual 
conditions  must  exist  in  order  to  make  these  particular 
effects  strikingly  obvious ;  if  these  exceptional  conditions  are 
observed,  however,  there  is  no  difficulty  in  verifying  the 
results  of  the  theory. 

Again,  if  the  opening  through  which  the  waves  come  is 
less  than  a  wavelength,  there  is  no  point,  such  as  p%,  whose 
distance  from  one  edge  of  the  aperture  is  a  whole  wave- 
length greater  than  that  from  the  other;  consequently  the 
most  striking  peculiarity  of  the  effect,  namely,  the  regions 
of  quiescence,  is  wholly  wanting.  In  the  case  of  sound  the 
average  lengths  of  the  waves  corresponding  to  the  voice  of 
a  man  is  not  far  from  eight  feet  and  of  a  woman's  voice 
about  half  as  great;  hence  we  ought  to  have  apertures  a 
number  of  feet  across  in  order  to  produce  the  required  effect. 
But  as  soon  as  our  apertures  become  as  great  as  this,  they  are 
of  the  same  order  of  magnitude  as  are  the  enclosures  in 
which  our  observations  are  made,  so  that  waves  reflected  from 
the  surrounding  walls  entirely  mask  the  effects  sought,  at 
least  to  one  not  guided  by  theory  in  his  search.  Thus  it 
is  not  surprising  that  this  particular  class  of  phenomena  in 
sound  waves  failed  of  recognition  until  after  their  discovery 
in  the  case  of  light. 

Finally,  if  the  aperture  is  many  times  larger  than  the 
length  of  the  wave,  the  nearer  and  stronger  images  will  lie 
so  close  to  the  primary  as  to  escape,  perhaps,  our  powers  of 
perception.  This  is  generally  the  case  with  light  where  the 
waves  are  very  short. 


INTERFERENCE  35 

Since  to  our  eyes  a  star  appears  as  a  point  of  light,  not  as 
a  spot  of  light  surrounded  by  a  series  of  concentric  circles, 
notwithstanding  that  the  waves  from  the  source  are  in  fact 
bounded  by  the  circular  inner  edge  of  the  iris,  we  must  con- 
clude that  light  waves  are  very  many  times  shorter  than  the 
diameter  of  the  pupil  of  the  eye,  or,  in  other  words,  that 
very  many  wavelengths  would  be  comprised  in  the  length  of 
about  one-eighth  of  an  inch.  According  to  the  theory,  the 
separation  of  the  images  increases  directly  as  the  aperture 
decreases;  consequently,  if  we  restrict  the  aperture  of  the 
eye,  we  ought  finally  to  secure  a  perceptible  separation  of 
the  secondary  images.  Thus,  if  we  look  through  a  needle 
hole  in  a  card  at  a  very  bright  point,  such  as  a  distant  elec- 
tric light,  or,  more  conveniently,,  a  bright  bead  or  ther- 
mometer bulb  in  sunshine,  the  central  disk  and  luminous 
surrounding  rings  become  obvious  at  once.  Moreover,  in 
accordance  with  theory,  the  smaller  the  hole  in  the  card  the 
larger  the  disk  and  its  concentric  rings,  though  of  course 
fainter  as  less  light  is  allowed  to  enter  the  eye.  If  the 
luminous  point,  which  we  may  conveniently  call  an  artificial 
star,  is  very  brilliant,  two  or  three  or  even  more  rings  may 
be  seen ;  but  if  a  less  brilliant  source  is  employed,  the  outer 
rings,  which  very  rapidly  decrease  in  brightness,  will  be  too 
faint  to  be  seen. 

The  modifications  in  the  image  produced  by  a  concave 
wave-surface  passing  through  two  small  holes  in  an  opaque 
screen  are  particularly  interesting,  because  we  shall  find  that 
they  yield  a  ready  means  of  measuring  the  value  of  the  length 
of  the  light  waves. 

Suppose  ab  and  afb',  Figure  15,  to  represent  two  circular 
holes  through  which  the  wave-front  whose  centre  is  at  p 
passes.  Then  the  light  which  passes  through  ab  forms  an 
image  in  the  region  about  p,  consisting  of  a  central  disk  and 
concentric  rings,  as  we  have  seen;  so,  too,  the  light  which 
passes  through  a'b'  forms  a  similar  image  at  the  same  place. 
If  the  waves  which  pass  through  the  former  opening  had  no 
definite  relations  to  those  which  pass  through  the  latter,  as 


36  LIGHT 

would  be  the  case,  for  example,  if  the  waves  through  ab 
came  from  one  artificial  star  and  those  through  afb'  from 
another,  the  effect  would  be  simply  to  make  the  image  twice 
as  bright,  since  twice  as  much  light  falls  upon  that  portion 
of  the  screen.  But  if  the  waves  at  the  two  apertures  are 
congruent,  which  would  be  the  case  if  they  came  from  the 
same  source  and  had  been  subject  to  the  same  conditions 
before  reaching  the  screen,  the  image  will  be  profoundly 
changed.  In  this  case  we  should  s,till  find  the  disk  and 
concentric  rings,  but  they  would  be  crossed  by  a  series  of 
fine,  dark,  nearly  straight  lines  perpendicular  in  direction  to 
the  line  joining  the  centres  of  the  holes.  That  this  must  be 


r 
* 


FIGURE  15. 

so  follows  from  simple  considerations  of  the  principles  in- 
volved. The  distance  from  the  centre  to  the  first  dark  ring, 
which  we  may  call  the  radius  of  the  primary  image,  is,  as  has 
been  shown  in  the  discussion  connected  with  Figure  14, 
nearly  1.2  times  the  distance  of  a  wavelength  multiplied  by 
the  ratio  of  ap  to  ab.  But  starting  from  p,  long  before  we 
come  to  the  point  p^  we  come  to  a  region  pr,  such  that  the 
difference  of  its  distance  from  a  and  a'  is  half  a  wavelength, 
and  hence  the  effect  produced  at  p1  by  the  portion  of  the 
wave  a  is  destroyed  by  that  produced  by  a'.  If,  however, 
the  centre  a  is  thus  neutralized  by  the  centre  a',  the  effect 
of  every  other  point  within  the  region  ab  will  be  neutralized 
by  the  corresponding  point  in  a'b1  ;  consequently  we  shall 


INTERFERENCE  37 

have  no  light  at  p'.  But  this  reasoning  is  equally  applicable 
to  a  point  p11,  which  is  at  three-halves  wavelengths  greater 
distance  from  a'  than  from  #,  and  also  when  this  difference  is 
any  odd  number  of  half  wavelengths.  Hence  there  will  be 
a  series  of  dark  lines  such  as  described.  It  will  be  observed 
that  the  diameter  of  the  image  is  inversely  as  the  diameter 
of  the  apertures,  but  that  the  separation  of  the  dark  lines  is 
inversely  as  the  distance  between  the  centres  of  the  holes,  the 
two  features  of  the  phenomenon  thus  depending  on  entirely 
different  elements.  This  interesting  phenomenon  can  be  easily 
seen  and  the  theory  verified  by  making  two  needle  holes  in  a 
card  at  a  distance  considerably  less  than  the  diameter  of  the 
pupil  of  the  eye,  and  looking  through  them  at  an  artificial 
star.  Figure  16  is  a  photographic  reproduction  of  this 
appearance  in  which  an  ordinary  camera 
replaced  the  eye  and  a  sensitive  plate 
the  retina. 

The  chief  interest  of  the  experiment, 
as  noted  above,  lies  in  the  fact  that  it 

enables  us   to  measure  the  length  of 

,.    ,  ,  .,,  .  .  .  .  FIGURE  16. 

light  waves  with  surprising  precision 

with  no  other  apparatus  than  a  finely  divided  scale,  say  to 
hundredths  of  an  inch,  which  may  be  obtained  at  any  hard- 
ware store.  Even  a  carpenter's  rule,  divided  to  sixteenths 
of  an  inch,  will  enable  us  to  measure  waves  of  light  by  this 
means  with  more  ease  than  we  can  measure  water  waves  to 
the  same  degree  of  precision.  As  this  value  is  the  funda- 
mental unit  of  the  whole  theory  of  light  and  of  nearly  every 
phenomenon  which  is  studied  in  the  following  pages,  the 
reader  is  strongly  recommended  to  make  the  measurement 
for  himself. 

To  do  so,  make  a  series  of  pairs  of  fine  holes  in  a  card 
with  a  needle  and  look  through  them  at  an  artificial  star.  If 
the  pair  of  holes  is  separated  by  an  interval  of  a  twelfth  of 
an  inch  or  more,  no  lines  across  the  disk-image  will  be  seen  ; 
if  the  interval  is  a  twentieth  of  an  inch  or  less,  the  lines 
become  very  distinct.  Select  a  pair  of  holes  at  such  a  dis- 


38  LIGHT 

tance  that  the  interference  lines  can  he  just  seen,  and  measure 
the  distance  between  the  holes  with  the  scale.  It  will  be 
found  easy  to  measure  this  interval  to  much  closer  than  one 
two-hundredth  part  of  an  inch,  if  the  scale  is  divided  to 
hundredths  of  an  inch,  and  to  a  hundred-and-sixtieth  part  by 
estimating  tenths  of  the  interval  of  an  ordinary  carpenter's 
rule.  Call  this  distance  D.  We  require  now  to  determine 
the  ratio  of  the  distance  p'p"  (or  its  equal  ppi)  to  the  distance 
ap,  referring  to  Figure  15.  Neither  of  these  quantities  can 
be  determined  readily  by  itself,  but  the  ratio  can  be  easily 
found  as  follows :  Draw  a  series  of  parallel  lines  on  white 
paper  in  ink,  making  the  width  of  the  lines  approximately 
equal  to  the  distance  between  them.  Fasten  this  paper  to 
the  wall  and  find  the  distance  at  which  they  can  be  just 
seen  as  separate  lines,  the  eye  being  aided  by  the  proper 
spectacle  glass  if  necessary.  The  ratio  of  the  distance  apart 
of  the  lines  to  this  distance  is  the  ratio  sought,  since  by 
supposition  the  system  of  interference  lines  was  also  just 
visible  as  separate  lines.  Finally,  as  appears  from  Figure  15, 
the  wavelength  of  light  is  D  times  this  ratio. 

For  examples  we  may  cite  the  following :  It  was  found  by 
an  observer  that,  looking  through  certain  pairs  of  holes  at 
a  thermometer  bulb  in  sunshine,  no  lines  could  be  seen 
through  the  first,  very  distinct  lines  through  the  second,  and 
the  finest  possible  lines  through  the  third.  A  measurement 
of  the  intervals  gave  0.08,  0.05,  and  0.065  of  an  inch,  respec- 
tively. For  that  observer's  eye,  then,  D  equals  0.065  of  an 
inch.  Then  five  parallel  lines  were  drawn  on  paper  which, 
it  was  found,  could  be  just  seen  as  separate  lines  at  a  dis- 
tance of  twenty-seven  feet.  The  measurement  of  the  group 
of  lines  showed  their  width  to  be  0.48  of  an  inch,  and  con- 
sequently 0,12  of  an  inch  as  the  interval  from  one  line  to 
the  next.  The  required  ratio  is  therefore  0.12  of  an  inch  to 
27  feet,  which  is  2700,  and  the  wavelength  of  light  is  0.067  of 
an  inch  divided  by  2700,  which  is  almost  exactly  1/40000 
of  an  inch.  A  second  observation,  in  which  only  a  car- 
penter's rule  was  employed  and  all  the  quantities  involved 


WAVELENGTH  39 

were  redetermined,  gave  1.05/16  of  an  inch  for  Z>,  and 
1/20  of  an  inch  for  the  distance  between  the  lines  on  the 
paper  which  could  be  seen  as  vanishingly  fine  from  a  distance 
of  twelve  feet.  These  data  by  the  same  process  of  reduction 
give  1/2880  of  the  distance  between  the  two  holes  for  the 
length  of  the  wave,  or  1/45000  of  an  inch.  The  estimated 
value  of  D  means,  of  course,  that  the  distance* from  one  hole 
to  the  other  is  a  little  greater  than  one-sixteenth  of  an  inch, 
but  not  so  much  as  a  tenth  greater. 

Of  these  values,  not  very  discordant  in  themselves,  the 
second  is  the  better,  and  is  very  close  indeed  to  the  true 
value,  which  for  white  light  may  be  regarded  as  equal  to 
1/45200  of  an  inch.  In  this  better  determination  the  paper 
with  parallel  lines  to  find  the  resolving  power  of  the  eye  was 
put  in  full  sunshine,  thus  approximating  more  closely  to  the 
condition  of  very  bright  and  very  dark  lines,  such  as  those 
of  the  interference  phenomenon. 

If  three  holes  are  made  in  the  card,  there  must  be  a  set  of 
dark  lines  for  each  pair,  that  is,  three  sets  of  lines.  Thus 
the  general  effect  will  be  that  of  three  groups  of  lines  cross- 
ing each  other  at  the  same  angles  as  those  of  the  triangle 
formed  by  the  centres  of  the  three  holes.  This  is  not  true 
with  absolute  exactness,  because  a  region  where  the  effects  of 
two  of  the  apertures  destroy  each  other  is  still  illuminated  by 
light  from  the  third,  so  that  extremely  beautiful  and  compli- 
cated patterns  may  occur,  although  the  general  configuration 
is  given  by  the  simple  rule  of  a  set  of  interference  lines  for 
each  pair  of  holes.  For  example,  three  holes  at  equal  dis- 
tances from  each  other  yield  a  series  of  hexagonal  images 
arranged  as  are  the  cells  in  a  honeycomb,  although  to  the 
eye  it  would  appear  simply  as  three  systems  of  equidistant 
lines  crossing  the  disk  at  mutual  angles  of  60°.  More 
important  is  the  deduction  which  we  can  make  from  the 
indifference  as  regards  the  position  of  the  holes  before  the 
eye,  provided  only  that  the  light  from  no  one  of  them  is  cut 
off  from  the  retina  by  the  iris.  That  this  position  is  in- 
essential follows  from  the  discussion  of  Figure  15,  where  the 


40  LIGHT 

iris  is  quite  left  out  of  the  question.  Hence  we  might  have 
two  systems  of  holes  of  exactly  the  same  size  and  configura- 
tion so  far  apart  that  light  from  one  system  would  not  visibly 
modify  that  from  the  other;  in  this  case  the  effect  would 
be  only  to  double  the  quantity  of  light  which  forms  the 
image.  But  it  is  not  difficult  to  see  that  the  second  system 
of  holes  need  not  be  remote  from  the  first  if  there  are  no 
new  distances  introduced  except  multiples  of  the  original 
distances.  Hence,  if  a  piece  of  perforated  cardboard,  such 
as  is  used  for  worsted  embroidery,  is  held  before  the  eye 
while  looking  at  the  artificial  star,  the  effect,  though  much 
brighter  and  the  separate  images  smaller,  is  the  same  in  kind 
as  though  only  four  of  the  holes  were  transparent.  The 
phenomena  presented  by  small  luminous  points  seen  through 
fine  and  regularty  woven  fabrics,  such  as  silk,  lawn,  bolting- 
cloths,  wire  cloth,  etc.,  are  of  this  kind.  Many  feathers  will 
exhibit  beautiful  effects  in  this  way.  A  modification  of  the 
phenomenon  may  be  made  by  holding  the  perforated  card- 
board in  front  of  the  objective  of  a  telescope  directed  toward 
a  bright  star.  In  this  case  we  virtually  increase  the  dimen- 
sions of  the  eye  and  can  use  correspondingly  coarse  structures 
in  the  perforated  screen.  Many  of  these  figures  are  of  sur- 
prising beauty,  and  in  all  of  those  where  the  image  is  bright 
and  sufficiently  large,  brilliant  colors  are  seen.  The  explana- 
tion of  these  colors  we  shall  find  in  the  next  chapter. 


CHAPTER  IV 

DISPERSION  —  CHROMATIC   EFFECTS  OF   DIFFERING 
WAVELENGTHS  —  COLORS   OF  THIN  PLATES 

HERETOFORE  we  have  tacitly  assumed  that  the  waves 
given  out  by  a  luminous  point  are  all  alike,  under  which 
assumption  we  have  explained  by  means  of  Huyghens's  prin- 
ciple the  phenomena  of  reflection,  refraction,  and  interfer- 
ence. But  in  general  waves  of  a  very  great  range  of  length 
are  emitted  by  such  a  point.  This  fact  changes  nothing  in 
the  reasoning  concerning  reflection  from  large  surfaces ;  but 
interference,  which  depends  upon  the  length  of  the  waves, 
becomes  obviously  more  complicated  in  such  cases ;  and,  what 
is  not  obvious  from  that  which  goes  before,  the  phenomena 
of  refraction  are  also  modified.  To  show  this,  vary  the 
experiment  described  on  page  13  with  a  wedge-shaped  glass, 
or  prism.  If  sunlight  is  transmitted  through  such  a  body 
and  received  on  a  distant  white  screen,  it  will  be  found  that 
instead  of  a  white  spot  marking  the  region  where  the  de- 
flected light  falls,  we  shall  have  a  brilliantly  colored  strip, 
the  end  nearest  the  place  where  the  undeviated  light  would 
fall  being  red,  and  that  most  remote  from  this  point  violet. 
The  intermediate  colors,  taken  in  order  from  the  red,  are 
orange,  yellow,  green,  green-blue,  blue.  The  change  from 
one  of  these  hues  to  the  next  is  absolutely  continuous,  so 
that  the  number  of  colors  is  limited  only  by  the  number 
of  names  at  our  command  for  designating  them.  Since  the 
change  in  direction  of  propagation  of  the  waves  depends  only 
on  their  less  velocity  in  glass  than  in  air,  we  conclude  that 
those  waves  which  produce  the  sensation  of  red  move  less 
slowly  than  those  which  give  rise  to  the  sensation  of  orange, 


42  LIGHT 

and  so  on.  This  separation  according  to  wave  slowness  in 
any  refracting  substance  is  called  dispersion,  and  the  result- 
ing colored  strip  is  called  a  spectrum.  Sir  Isaac  Newton  was 
the  first  one  to  investigate  this  phenomenon  in  a  scientific 
manner,  and  to  fix  its  terminology,  though  he  used  the  color 
names  green,  blue,  indigo-blue,  and  violet,  instead  of  green, 
green-blue,  blue,  and  violet,  which  modern  writers  have  found 
more  appropriate.  Newton's  most  important  discovery  was 
that  after  the  light  is  thus  modified  any  one  color  suffers  no 
further  change  by  passing  through  another  prism.  His  con- 
clusion was  that  ordinary  white  light  is  compound  and  made 
up  of  an  indefinite  number  of  hues,  of  which  seven  are 
recognized  by  familiar  names.  He  confirmed  this  deduction 
by  showing  that  if  these  differently  colored  lights  were 
united,  either  by  allowing  them  to  fall  upon  a  concave  mirror 
and  reflecting  them  to  a  common  point,  or  by  passing  the 
decomposed  light  through  a  similar  prism  turned  in  an  oppo- 
site direction,  the  result  would  be  white  light  like  that  from 
the  original  source.  Thus  both  analysis  and  synthesis  were 
employed  at  his  hands  to  demonstrate  the  composite  nature 
of  white  light. 

Newton  also  found  that  like  prisms  of  different  substances 
would  produce  quite  dissimilar  amounts  of  dispersion ;  but, 
fortunately  for  the  development  of  practical  optics,  his  con- 
clusion, that  the  phenomenon  of  dispersion  (which  should 
be  looked  upon  as  a  secondary  phenomenon  of  refraction 
on  account  of  its  relative  minuteness)  increases  directly  as 
the  refractive  power,  was  an  error. 

From  the  Newtonian  standpoint  the  explanation  of  dis- 
persion was  found  in  the  assumption  of  as  many  different 
kinds  of  small  corpuscles  as  there  are  different  hues ;  in  the 
wave  theory  the  explanation  can  be  found  only  in  attributing 
the  differences  of  velocity  in  the  material  of  the  prism  and 
the  corresponding  colors  to  differing  wavelengths.  That  the 
latter  explanation  is  adequate  and  in  accordance  with  the 
facts,  we  may  prove  by  means  of  the  perforated  card  and 
the  artificial  star.  This  may  be  done  in  various  ways.  We 


DISPERSION  43 

may  form  a  spectrum  by  a  prism  held  in  the  sunlight,  and 
observe  the  artificial  star  produced  when  the  thermometer 
bulb  is  moved  from  one  end  to  the  other  of  the  spectrum. 
If  the  star  which  is  of  the  color  of  that  portion  of  the  spec- 
trum in  which  it  is  immersed,  is  observed  through  the  two 
needle  holes,  it  will  be  seen  at  once  that  the  interference 
lines  are  furthest  apart  in  the  red  and  continuously  approach 
as  the  star  changes  successively  to  orange,  yellow,  green, 
etc.  Or  we  may  place  the  thermometer  bulb  in  the  sunshine 
and  look  at  it  through  a  prism ;  the  artificial  star  will  then 
appear  as  a  fine  linear  spectrum.  If,  then,  the  screen  with 
the  two  needle  holes  is  placed  close  to  the  eye,  between  it 
and  the  prism,  with  the  line  joining  the  holes  at  right  angles 
to  the  spectrum,  the  latter  will  appear  broadened  and  traversed 
by  a  series  of  fine  longitudinal  lines,  which  are  the  interfer- 
ence lines.  It  will  be  easy  to  recognize  that  these  lines  are 
most  widely  separated  at  the  red  end  of  the  spectrum.  The 
second  method,  though  not  quite  as  perspicuous  as  the  first, 
has  the  advantage  of  requiring  only  a  very  small  prism ;  the 
first  requires  a  large  prism  in  order  to  have  the  star  suffi- 
ciently bright. 

A  third  method,  which  enables  us  to  dispense  altogether 
with  a  prism,  though  otherwise  inferior,  is  to  employ  colored 
glasses  in  conjunction  with  the  pierced  card.  Thus  one 
observer  found  that  looking  through  the  card  and  red  glass 
he  could  see  the  interference  lines  when  the  holes  were  one- 
twelfth  of  an  inch  apart,  with  yellow  glass  it  required  a  dis- 
tance not  much  greater  than  one-sixteenth,  and  with  blue 
glass  he  found  it  necessary  to  have  the  holes  within  one- 
twentieth  of  an  inch  of  each  other. 

The  conclusions  from  these  experiments  are,  first,  that  the 
velocity  of  propagation  of  light  waves  in  glass  decreases  con- 
tinuously with  decrease  of  wavelength;  second,  that  waves 
of  different  lengths  falling  on  the  retina  produce  different 
color  sensations,  the  longest  waves  awakening  the  sensation 
of  red  and  the  shortest  that  of  violet;  and  third,  as  appears 
from  the  quantities  given  in  the  third  experiment,  that  the 


44  LIGHT 

violet  waves  cannot  be  very  much  greater  or  very  much  less 
than  half  the  length  of  the  red  waves.  As  regards  this  last 
conclusion,  we  may  add  that  it  is  impossible  to  state  the  limit- 
ing values  of  the  waves  which  can  stimulate  the  retina,  but 
without  extraordinary  precautions  we  may  fairly  assume 
1/2TOOO  of  an  inch  for  the  extreme  red  and  1/52000  of  an 
inch  for  the  extreme  violet;  thus  the  ratio  of  the  longest  to 
the  shortest  is  about  1.94  to  1. 

To  this  enlarged  view  of  the  nature  of  white  light  it  is 
easy  to  adapt  the  conclusions  of  the  preceding  chapters.  As 
regards  the  phenomena  of  reflection,  nothing  is  to  be  added 
to  what  appears  in  the  first  chapter;  but  refraction  at  a  plane 
surface,  as  in  Figure  5,  forms,  instead  of  a  virtual  image  of 
the  source,  a  virtual  spectrum  of  which  the  violet  end  is 
nearer  the  refracting  surface.  The  action  of  a  plate  is  to 
form  a  virtual  spectrum  with  the  violet  end  nearer  the  plate, 
but  this  effect  can  be  seen  only  in  the  case  of  a  very  thick 
plate,  and  ordinarily  when  the  object  is  looked  at  obliquely. 
Thus  a  white  pebble  seen  vertically  downward  through  deep 
water  still  appears  white,  but  if  seen  away  from  the  plumb- 
line,  colors  of  the  spectrum  become  very  obvious.  So  a  prism 
instead  of  forming  an  image  of  the  source  displaced  toward 
the  thin  edge  of  the  prism  and  approached  by  a  fraction  of 
the  thickness  of  the  prism,  as  appears  from  the  consideration 
of  Figure  6,  in  reality  forms  a  spectrum  with  the  blue  end 
more  displaced  than  the  red  end,  and  also  brought  a  very 
little  nearer. 

The  description  of  interference  phenomena  produced  by 
limited  wave-surfaces  is  quite  easily  extended  to  include 
the  color  effects  due  to  varying  wavelengths.  It  is  only 
necessary  to  remember  that  the  images  thus  produced  have 
dimensions  in  direct  ratio  to  the  wavelength  of  the  light 
which  goes  to  form  them.  Accordingly,  the  image  of  a  white 
artificial  star  consists  of  a  disk  of  which  the  centre  is  white, 
since  all  waves  are  there  represented,  but  with  a  red  or 
orange  margin ;  the  rings  immediately  surrounding  the  disk 
are  blue  on  the  inner  side  and  red  externally.  Though  this 


INTERFERENCE  FIGURES  45 

is  the  true  description  of  the  image,  it  is  impossible  to  recog- 
nize it  as  such  because  of  the  limits  of  our  perceptive  facul- 
ties, for  if  the  aperture  is  made  large  so  as  to  render  the 
colors  brilliant,  the  disk  and  rings  will  be  too  small  to  be 
perceived  in  their  details ;  on  the  other  hand,  if  the  opening 
is  very  small,  so  that  the  various  features  of  the  image  are 
large  enough  to  be  obvious,  the  light  will  be  so  faint  that 
the  colors  are  unrecognizable,  just  as  we  are  unable  to  name 
colors  in  even  quite  bright  moonlight.  With  two  or  more 
apertures,  however,  we  may  succeed  better.  In  the  case  of 
two  holes  we  may  describe  the  resulting  image  as  composed 
of  an  indefinite  number  of  superimposed  images,  all  having 
the  same  centre,  but  of  sizes  increasing  regularly  with  the 
color  from  violet  to  red.  Hence  the  middle  band  would  be 
white,  flanked  on  either  sicle  by  bands  of  which  the  inner 
edge  is  blue  and  the  outer  red.  As  we  go  outward  from 
the  centre,  the  chromatic  separation  will  become  greater  and 
greater,  until  finally  we  reach  a  point  beyond  which  every 
color,  though  of  course  not  every  wavelength,  will  be  repre- 
sented at  all  points.  In  all  such  regions  the  image  will 
appear  white,  and  the  immediate  effects  of  interference  will 
vanish.  This  limit  in  the  case  of  white  light  is  found  to  be 
reached  in  the  region  of  the  eighth  or  ninth  band,  so  that  in 
white  light  we  can  never  see  more  than  seventeen  or  nine- 
teen bands,  even  under  favorable  circumstances,  though  with 
light  of  a  single  color  it  is  sometimes  possible  to  see  many 
thousands. 

In  the  experiment  with  two  apertures  the  chief  difficulty 
in  seeing  the  colors  lies  in  the  fineness  of  the  lines  and  the 
faintness  of  the  illumination,  as  indicated  above;  but  with 
holes  a  thirtieth  of  an  inch  apart  or  less,  the  lines  are  suffi- 
ciently coarse,  though  if  many  bands  are  to  be  seen  the  holes 
must  be  very  small.  We  have  learned,  however,  that  greatly 
increasing  the  number  of  apertures,  provided  that  they  are 
systematically  arranged,  produces  little  change  except  in- 
creased brightness ;  hence  by  this  means  the  color  effects  may 
be  made  very  marked.  An  artificial  star  seen  through  fine, 


46  LIGHT 

uniformly  woven  silk  or  through  the  web  of  a  uniform 
feather  will  show  very  beautifully  colored  figures.  As  both 
nature  and  art  present  us  with  numerous  examples  of  such 
structures,  we  could  hardly  fail  to  be  familiar  with  this  class 
of  phenomena  were  it  not  that  we  very  rarely  look  through 
such  bodies,  but  only  at  them.  Perhaps  the  commonest 
example  where  conditions  both  of  regularity  of  structure  of 
the  screen  and  smallness  of  the  source  of  light  are  met  is  that 
of  an  electric  light  seen  through  a  silk  umbrella.  The  type 
of  figure  here  presented  is  that  of  an  artificial  star  seen 
through  three  holes  at  the  vertices  of  a  right-angled  triangle. 
But  there  is  a  very  large  class  of  examples  of  such  color 
effects  produced  in  a  different  manner  yet  with  precisely 
the  same  theoretical  explanation.  We  have  seen,  on  page  9, 
that  optically  a  small  mirror  is  exactly  the  same  as  a  hole  of 
its  own  size  through  which  we  can  see  a  perverted  image 
of  all  that  lies  in  front  of  its  plane.  Consequently,  looking 
toward  a  number  of  small  mirrors  is  optically  the  same  as 
looking  through  a  number  of  like  apertures,  except  that  the 
source  or  sources  of  light  appear  to  be  in  directions  other 
than  their  real  ones.  In  its  action  with  respect  to  light 
reflected  from  it  a  piece  of  polished  glass  or  metal  with  close, 
uniformly  distributed  grooves  would  resemble  that  of  a  series 
of  fine  slits  in  an  opaque  screen  with  respect  to  transmitted 
light.  When  held  close  to  the  eye  a  small  source  of  light 
would  appear  flanked  on  either  side  by  a  series  of  colored 
images;  at  a  distance  from  the  eye  it  would  appear  of  the 
color  proper  to  the  kind  of  waves  that  are  not  mutually 
self-destructive  in  the  direction  of  the  eye.  This  peculiar 
effect,  which  depends  upon  the  direction  in  which  the  grooved 
reflector  is  observed  and  upon  the  direction  of  the  source  of 
light,  is  called  iridescence.  Mother-of-pearl,  which  is  com- 
posed of  smooth  reflective  layers,  shows  this  property  in  a 
marked  degree,  on  account  of  its  peculiar  structure.  That  it 
is  due  to  the  structure  alone  may  be  proved  by  taking  a  print 
of  a  piece  of  bright  mother-of-pearl  on  white  wax,  when  it 
will  be  found  that  the  surface  of  the  wax  also  shows  irides- 


COLORS  BY  INTERFERENCE  47 

cent  colors.  Feathers  possessing  at  the  same  time  regular 
structure  and  brilliant  lustre  exhibit  the  same  phenomenon. 
Most  of  the  beauty  of  peacock's  feathers,  and  all  of  that 
of  the  iridescent  feathers  of  the  male  birds  of  the  turkey, 
pigeon,  and  humming-bird  families  are  thus  explained.  If 
the  source  of  light  is  very  large  or  if  the  structure  of  the 
reflecting  surface  is  somewhat  irregular,  the  colors  are  less 
pronounced  or  may  be  wholly  wanting,  the  only  effect  being 
a  remarkable  change  of  lustre  with  varying  obliquity.  The 
sheen  of  white  mother-of-pearl,  of  the  gem  called  cat's -eye, 
as  well  as  that  of  satin  spar,  is  of  this  kind.  The  variety 
of  feldspar  known  as  labradorite  has  such  regularity  of 
fibrous  structure  that  a  polished  surface  illuminated  by  a 
source  of  light  of  moderate  extent  will  show  most  vivid 
colors. 

There  is  a  difference  in  the  arrangement  of  colors  pro- 
duced by  a  prism  and  by  interference,  which  is  characteristic, 
and  worthy  of  note.  In  the  former  case  the  blue  end  of  the 
spectrum  is  further  away  from  the  undeviated  position  of  the 
light  than  the  red,  while  in  the  latter  case  this  rule  is  re- 
versed. This  difference  may  enable  one  to  distinguish  in 
some  cases  the  origin  of  a  color.  For  example,  the  corona 
seen  about  the  moon  when  covered  by  a  very  light  cloud  is 
blue  within  and  red  on  the  edge  more  remote  from  the  moon ; 
on  the  other  hand,  the  large  colored  circle  which  is  called  a 
halo,  and  which  is  not  infrequently  seen  around  the  sun  or 
moon  in  cold  weather,  has  a  red  border  on  the  inner  edge  and 
a  blue  one  on  the  outer.  According  to  the  foregoing  principle, 
we  ought  to  conclude  that  the  first  is  due  to  interference  and 
the  second  to  refraction.  This  is  really  the  case,  for  the 
corona  is  produced  by  the  diffractive  action  of  small  and 
uniform  drops  of  water  (which  act  in  the  same  way  as  a 
number  of  holes  of  the  same  diameter),  and  the  halo  is  due 
to  the  refraction  of  the  light  by  small  sixty-degree  prisms  of 
ice  suspended  in  the  atmosphere. 

There  is  another  method  of  securing  interference  of  light 
which,  though  it  has  no  bearing  on  the  .theory  of  vision  or 


48  LIGHT 

on  the  theories  of  optical  instruments,  is  of  such  common 
occurrence  that  we  are  justified  in  entering  upon  its  treat- 
ment here.  Instead  of  using  two  portions  of  the  same  wave- 
surface,  we  may  divide  a  system  of  waves  into  two  separated 
systems  and  allow  them,  after  having  traversed  different 
paths,  to  reunite.  There  are  numerous  ways  of  doing  this, 
but  we  shall  confine  our  attention  for  the  present  to  the  most 
common  method. 

Suppose  a  system  of  light  waves  to  fall  upon  a  very  thin 
plate  of  some  transparent  substance,  water,  for  example ;   the 
waves  would  be  reflected  in  part  from  the  first  surface,  but 
the  greater  portion  would  pass  on  through  the  plate  until  it 
fell  on  the  second  surface,  where  another  part  would  suffer 
reflection,   and,  with   slight'  loss   at  the   emergent  surface, 
re-enter   the  air.     This  second   system  of   waves   would  be 
obviously  behind    the   first  by  twice   the   thickness  of  the 
plate,  if  the  reflection  were  nearly  normal.    Suppose  now  that 
the  plate  has  a  thickness  of  one-fourth  of  a  wavelength  of  red 
light,  then  the  two  systems  would  be  mutually  destructive  as 
regards  these   particular  waves,  so  that  their  effect   on  the 
retina  would  be  that  of  white  light  minus  red,  or,  as  we  shall 
see  in  the  second  part  of  this  book,  it  would  appear  green- 
blue.     If  the  plate  were  a  little  thinner,  say  one-fourth  the 
yellow  wavelength,    this    color    would    be    absent    in    the 
combined  effect,  and  the  plate  would  appear  blue.     Again, 
suppose  the  plate  to  have  a  thickness  of  three-fourths  of  a 
wavelength  of  red  light,  then  the  retardation  of  the  waves 
reflected  from  the  back  surface  would  be  three-halves  of  a 
wavelength,  and  mutual  destruction  of  this  particular  wave- 
length would  ensue,  with  a  corresponding  modification  of  the 
color.     This  is  essentially  the  explanation  of  the  production 
of  colors  by  reflection  from  thin  plates,  such  as  soap-films  or 
thin  layers  of  oil  on  the  surface  of  water.     To  make  the  ex- 
planation quite  in  accordance  with  fact,  we  should  take  into 
account  the  curious  change   in   phase  that  waves   undergo 
when  reflected  at  the  passage  from  a  rarer  to  a  denser  medium, 
which   does   not   occur  in   reflection  under  reversed  condi- 


COLORS  OF  THIN  PLATES  49 

tions.  But  as  this  fact  does  not  cause  any  change  ordinarily 
perceptible,  we  may  ignore  it  here. 

There  is  one  notable  peculiarity  of  the  colors  produced  in 
this  way  which  is  of  great  interest,  not  only  because  it  affects 
the  whole  character  of  the  phenomena,  but  because  it  renders 
evidentafc  once  why  we  see  these  colors  in  very  thin  plates 
alone.  "Mt  depends  upon  the  fact  that  the  colors  are  not  those 
of  the  various  wavelengths,  as  in  the  case  of  the  prismatic 
spectrum,  or  even  those  of  the  sum  of  several  wavelengths,  as 
is  true  of  many  of  the  interference  phenomena  last  con- 
sidered, but  they  are  the  colors  proper  to  white  light  after 
being  deprived  of  one  or  more  definite  wavelengths,  j  Let  us 
suppose  that  we  have  a  plate  of  such  thickness  that  the 
yellow  is  eliminated  in  the  reflected  light ;  the  remainder  is 
blue,  since  these  two  hues  are  complementary.  But  if  the 
thickness  is  such  that  the  retardation  of  the  system  of  waves 
from  the  second  reflection  is  three-halves  of  a  wavelength  of 
red  light,  it  will  be  five-halves  of  a  wavelength  of  the  shorter 
green  waves ;  hence  the  color  will  be  white  minus  such  red 
and  green.  Now  red  and  green  light  combined  form  yellow; 
therefore  white  deprived  of  these  two  hues  will  be  blue  also, 
only  the  blue  is  notably  paler  than  that  from  the  thinner 
plate.  These  two  resultant  colors  are  called  blues  of  the 
first  and  of  the  second  order,  respectively.  An  extension 
of  the  reasoning  shows  that  a  still  thicker  plate  may  eliminate 
three  wavelengths,  which  would  yield  again  a  still  paler 
blue,  provided  that  the  three  combined  would  produce  a 
yellow  hue.  Should  the  increase  of  thickness  of  the  plate  be 
so  far  extended  that  certain  wavelengths  of  all  colors  are 
eliminated,  that  is,  that  a  series  of  colors  which  combined 
would  themselves  produce  white,  the  sum  of  the  remaining 
colors  would  be  also  white.  Thus  we  see  that  only  films  of 
a  very  few  wavelengths  in  thickness  can  produce  colors  by 
reflection,  and  also  that  there  is  a  limit  to  the  number  of 
orders  of  the  recurrent  colors. 

All  these  consequences  of  theory  may  be  observed  and 
verified  very  conveniently  by  soap-films,  either  by  watching  a 

4 


50  LIGHT 

moderate-sized  soap-bubble  protected  from  currents  of  air,  or, 
better  still,  by  dipping  a  wire  ring  two  or  three  inches  in 
diameter  into  a  solution  of  soap,  whence  it  can  be  removed 
with  a  film  across  it.  If  this  is  held  in  an  inclined  position 
where  the  reflection  of  the  sky  from  it  can  be  seen,  a  series 
of  colored  horizontal  bands  will  appear,  of  which  the  upper 
ones  correspond  to  the  thinner  plate.  It  will  be  observed  that 
the  hues  repeat  themselves,  but  grow  paler  with  each  repeti- 
tion. Sir  Isaac  Newton,  who  first  studied  the  phenomena  of 
thin  plates  and  their  causes,  devised  an  apparatus  for  show- 
ing all  the  colors  with  perfect  regularity.  It  consisted  of 
a  polished  spherical  glass  surface  of  very  small  curvature, 
pressed  against  a  flat  polished  plate.  When  skylight  is 
reflected  from  the  thin  plate  of  air  included  between  these 
two  bodies,  a  series  of  concentric  rings  may  be  seen,  of  which 
the  inner  ones  are  those  corresponding  to  the  thinner  portion 
and  possess  the  more  intense  coloring.  These  rings  are 
universally  known  as  Newton's  rings. 

Since  the  colors  produced  in  the  manner  described  are  to  be 
seen  by  looking  directly  at  the  reflecting  plate,  not  through 
or  beyond  as  in  the  cases  of  interference  previously  consid- 
ered, they  are  much  more  frequently  observed.  Besides  the 
cases  already  mentioned,  they  may  be  often  seen  in  partly 
fractured  glass  or  ice.  Quartz  and  other  crystals  not  un- 
commonly have  internal  fractures  which  show  brilliant  colors 
of  this  origin,  and  the  magnificent  play  of  colors  in  the 
precious  opal  is  explained  in  the  same  way.  The  surface  of 
glass  which  has  been  long  exposed  to  -dampness  undergoes  a 
change  which  gives  the  effect  of  a  thin  layer  of  transparent 
refracting  substance  superimposed  upon  it.  In  the  famous 
antique  glasses  of  Cyprus  the  layers  seem  to  be  multiple  and 
the  effects  are  wonderfully  brilliant.  This  is  imitated  with  a 
certain  amount  of  success  in  modern  glass,  by  exposing  the 
material  to  corrosive  vapors  at  high  temperatures. 

If  a  piece  of  polished  steel  is  exposed  to  the  air  when 
heated,  the  surface  is  oxidized  to  a  varying  degree  of  depth, 
depending  primarily  on  the  temperature.  This  thin  sheet  of 


COLORS   OF  THIN  PLATES  51 

oxide  reflects  light  as  a  thin  plate,  and  affords  a  valuable 
aid  to  the  mechanic  in  judging  the  proper  temperature  for 
tempering.  A  layer  of  such  thickness  as  to  eliminate  the 
green  waves,  thus  leaving  a  brilliant  purple,  is  often  used  as 
an  ornamental  finish  for  small  steel  work,  such  as  screws,  the 
hands  of  a  watch,  and  so  forth. 


CHAPTER  V 
THE    TELESCOPE 

THERE  is  no  instrument  which  has  done  so  much  to  widen 
the  scope  of  human  knowledge,  to  extend  our  notions  of  the 
universe,  and  to  stimulate  intellectual  activity,  as  has  the 
telescope.  Certainly  no  other  instrument  except  the  micro- 
scope could  for  a  moment  be  held  to  dispute  this  pre- 
eminence, but  the  former  instrument  is  incomparably  more 
interesting  in  its  history  in  the  same  degree  that  its  history 
is  more  simple  and  more  comprehensible.  To  trace  its 
development  from  a  curious  toy  in  the  hands  of  its  dis- 
coverer to  the  middle  of  this  century,  is  to  be  brought  into 
contact  with  most  of  the  great  philosophers  who  have  achieved 
eminence  in  physical  science,  from  the  time  of  the  Re- 
naissance. Galileo,  Torricelli,  Huyghens,  Cassini,  Keppler, 
Newton,  Euler,  Clairault,  the  Herschels,  father  and  son, 
Fraunhofer,  Gauss  —  these  form  only  a  portion  of  the  list  of 
great  names.  The  growth  of  the  telescope  toward  perfection 
has  constantly  carried  with  it  increase  of  precision  in  astron- 
omy, in  navigation,  and  in  all  branches  of  engineering.  It 
would  be  easy  to  show  that  even  pure  mathematics  would  be 
in  a  far  less  forward  state  had  there  been  no  problems  of 
astronomy  and  physics  which  were  first  suggested  by  the 
employment  of  this  instrument.  It  is  to  a  review  of  its 
history  that  this  chapter  is  to  be  devoted.  In  it  we  may 
hope  to  find  a  succinct  statement  of  the  origin  and  develop- 
ment of  this  potent  aid  in  the  study  of  nature,  a  mention  of 
some  of  the  more  important  achievements  depending  upon 
it,  and  a  sketch  of  its  gradual  improvement  to  the  magnificent 
and  complicated  instrument  which  constitutes  the  modern 


54  LIGHT 

equatorial.  Following  this  we  shall  strive  to  gain  an  idea 
of  the  imperfections  which  the  generations  of  ingenious 
artisans  have  had  to  contend  with  in  attaining  its  present 
degree  of  perfection,  and  possibly  to  forecast  in  a  manner 
what  the  future  may  bring  to  us  in  the  same  field. 

In  Chapter  II.  was  sketched  the  early  history  of  the  tele- 
scope, immediately  after  its  discovery  by  Lippershey  and  its 
application  to  scientific  investigation  by  Galileo.  That  this 
discovery  was  really  an  accident  we  may  be  quite  sure,  for 
not  only  was  there  no  developed  theory  of  optics  at  that 
time,  but  even  the  law  of  refraction,  which  lies  at  the  basis 
of  such  theory,  was  wholly  unknown.  So  also  it  seems  more 
than  probable  that  Galileo's  invention  was  empirical  and 
guided  by  somewhat  precise  information  originating  in  Hol- 
land, such  as  that  the  instrument  consisted  essentially  of  two 
lenses  of  which  one  was  a  magnifying  and  the  other  a  dimin- 
ishing lens ;  at  least,  that  Galileo's  telescope  was  like  that 
of  Lippershey;  that,  theroretically  considered,  it  is  not  as 
simple  as  one  made  of  two  magnifying  lenses,  as  is  evinced 
by  the  fact  that  the  first  philosopher  to  establish  an  approx- 
imate theory  of  optical  instruments  invented  the  latter  and 
prevailing  form ;  and  finally,  that  Galileo  published  no  con- 
tributions to  optics  —  all  these  reasons  together  seem  quite 
sufficient  to  enforce  such  a  belief.  But  in  any  case  Galileo's 
merit  is  in  no  wise  lessened  by  having  failed  to  do  what 
could  not  have  been  done  at  that  time  ;  and  to  him  more 
than  to  any  of  his  contemporaries  clearly  belongs  the  glory  of 
having  made  the  telescope  the  most  efficient  servant  of  scien- 
tific research  which  human  ingenuity  has  yet  invented. 

No  further  discoveries  of  great  moment  were  made  until 
over  a  generation  after  Galileo  proved  the  existence  of  spots 
on  the  sun,  in  1611.  This  cessation  of  activity  was  doubtless 
owing  to  the  difficulty  in  securing  telescopes  of  greater 
power  than  those  possessed  by  Galileo,  which  he  would 
hardly  have  left  until  their  capacity  for  discovery  had  been 
fully  exhausted  in  his  own  hands.  By  the  middle  of  the 
seventeenth  century,  however,  several  makers  of  lenses  had 


THE   TELESCOPE  55 

so  far  improved  the  methods  of  grinding  and  polishing  glass 
that  telescopes  notably  superior  in  efficiency  to  the  best  of 
Galileo's  were  procurable.  Of  these  Torricelli,  Divini,  and 
Campani;  the  French  Auzout,  who  constructed  a  telescope 
600  feet  long,  although  no  means  was  ever  devised  for  direct- 
ing such  an  enormous  instrument  toward  the  heavens ;  and 
above  all,  Huyghens,  have  won  distinction  as  telescope- 
makers.  The  last-named  philosopher,  by  means  of  a  tele- 
scope of  his  own  construction,  discovered,  in  1655,  the  largest 
satellite  of  Saturn,  thus  adding  a  fifth  member  to  the  list  of 
planetary  bodies  unknown  to  the  ancients.  But  his  most 
important  astronomical  discovery,  made  also  in  1655,  was  the 
nature  of  the  rings  of  Saturn.  This  object  had  greatly 
puzzled  Galileo,  to  whose  small  telescope  the  planet  appeared 
to  consist  of  a  larger  sphere  flanked  on  either  side  by  a 
smaller  one  ;  but  when,  in  the  course  of  the  orbital  motion  of 
Saturn,  the  rings  entirely  disappeared,  he  was  wholly  unable 
to  suggest  any  explanation  for  so  unexpected  a  phenomenon. 
The  planet  had  thus  presented  a  remarkable  problem  to  all 
astronomical  observers  for  more  than  forty  years,  and  the 
records  of  the  efforts  to  solve  it  during  that  interval  afford 
us  a  most  excellent  means  of  estimating  the  progress  in 
practical  optics.  Huyghens  announced  these  discoveries  in 
1656,  but  that  relating  to  the  ring  was  given  in  the  form  of 
an  anagram,  the  solution  of  which  was  first  published  in  1659. 
This  latter  discovery  was  contested  in  Italy  by  Davini,  but 
was  finally  confirmed  by  members  of  the  Florentine  Academy 
with  one  of  Davini's  own  telescopes. 

A  few  years  later  the  famous  astronomer  Cassini,  having 
come  to  Paris  from  Italy  as  Royal  Astronomer,  commenced  a 
series  of  brilliant  discoveries  with  telescopes  made  by  Cam- 
pani of  Rome.  With  these,  varying  in  length  from  35  to 
136  feet,  he  discovered  four  satellites  of  Saturn  in  addition 
to  the  one  discovered  by  Huyghens.  The  whole  number 
was  increased  by  Herschel's  discovery  of  two  smaller  ones  in 
1789,  a  hundred  and  five  years  after  Cassini's  last  discovery, 
and  again  by  Bond's  detection  of  an  eighth  in  1848.  The 


56  LIGHT 

Saturnian  system,  to  which  the  telescope  has  doubtless  been 
directed  more  frequently  than  to  anything  else,  thus  serves 
as  a  record  of  the  successive  improvements  of  the  telescope. 
Highly  significant  is  the  fact  that  the  discoveries  of  the 
eighteenth  century  were  made  with  a  reflecting  telescope, 
the  others  all  being  with  refracting  instruments. 

Cassini's  discovery  of  the  two  satellites  now  known  as 
Dione  and  Tethys,  in  1684,  was  not  accepted  as  conclusive 
until  long  afterward,  when  Pond,  in  1718,  with  a  telescope 
123  feet  long,  which  Huyghens  had  made  and  presented  to  the 
Royal  Society,  saw  all  five  of  the  satellites.  This  particular 
instrument  is  of  especial  interest,  because  it  is  the  only  one  of 
those  of  the  last  half  of  the  seventeenth  century  which  has 
been  carefully  compared  with  modern  instruments.  More- 
over, it  is  without  doubt  quite  equal  in  merit  to  the  best  of 
that  period.  But  we  find  that,  although  it  had  a  diameter  of 
six  inches,  its  performance  was  hardly  better  than  that  of  a 
good  modern  instrument  of  four  inches  in  diameter  and  per- 
haps four  and  a  half  feet  in  length,  while  in  regard  to  con- 
venience in  use  the  modern  compact  telescope  is  incomparably 
superior. 

Another  notable  discovery  of  this  period  was  that  of 
the  duplicity  of  the  rings  of  Saturn,  made  by  Cassini  in 
1675.  It  is  often  said,  apparently  after  a  statement  in 
Grant's  History  of  Astronomy,  that  this  discovery  was  made 
a  decade  earlier  by  the  brothers  Ball  in  England ;  but  it  is 
quite  impossible  to  find  authority  for  this  conclusion  from 
the  original  descriptions  of  their  observations  in  the  Trans- 
actions of  the  Royal  Society  where  they  were  published.  Of 
more  interest  to  us  is  the  fact  that  these  observations  appear 
to  have  been  made  with  a  telescope  38  feet  long,  of  English 
manufacture ;  if  this  is  true,  it  seems  to  be  the  first  notice  of 
the  existence  of  opticians  of  high  merit  in  that  country.  We 
must  regard  Cassini's  discovery  of  the  third  and  fourth  satel- 
lites of  Saturn,  however,  as  marking  the  very  furthest  reach 
of  the  old  form  of  telescope;  a  century  was  to  elapse  and  an 
entirely  new  form  was  to  be  developed  before  another  consid- 


THE   TELESCOPE 


57 


erable  addition  to  our  knowledge  of  the  aspect  of  the  heavenly 
bodies  was  to  be  made.  It  is  true  that  larger  telescopes  were 
made,  and  Huyghens  invented  a  means  by  which  they  could  be 
used  without  tubes ;  but  notwithstanding  this  improvement, 
they  proved  so  cumbersome  as  to  be  impracticable. 

The  older  opticians  had  found 
tha.t  if  they  attempted  to  increase 
the  diameter  of  a  telescope  they 
were  obliged  to  increase  its  length 
in  a  much  more  rapid  ratio  to 
retain  the  same  clearness  of  vision. 
The  reason  for  this  was  not  clearly 


FlGUBB   17. 

understood,  but  it  was  supposed  to  be  owing  to  the  fact  that 
a  wave-front  changed  in  curvature  by  passing  through  a 
spherical  surface  ceases  to  remain  strictly  spherical,  as  has 
been  shown  on  page  15  of  this  book.  This  deviation  in 
shape  of  the  refracted  wave  from  a  true  sphere  is  called 
spherical  aberration.  When  the  refracting  surfaces  are  large 


58  LIGHT 

and  of  considerable  curvature,  this  error  becomes  very  seri- 
ous; but  by  using  small  curvatures,  which  in  a  telescope 
obviously  corresponds  to  great  length,  the  effects  of  the  error 
can  be  made  insensible ;  hence  the  rapid  increase  of  ratio  of 
length  to  diameter  remains  unaccounted  for.  Newton's  dis- 
covery of  the  composite  nature  of  light  and  of  the  phenome- 
non of  dispersion  enabled  him  to  explain  the  true  cause  of 
indistinctness  in  short  telescopes,  namely,  that  the  refraction 
by  the  objective  varies  for  different  colors;  hence,  if  the 
ocular  is  placed  for  one  particular  color  it  will  not  be  in  the 
right  position  for  any  others,  whence  the  image  of  a  star 
or  planet  will  seem  to  be  surrounded  by  a  fringe  of  color. 
Newton  found  this  source  of  indistinctness  in  the  image, 
which  is  now  known  as  chromatic  aberration,  many  hundreds 
of  times  as  serious  as  the  spherical  aberration.  As  he  was 
persuaded  by  his  experiments  that  this  obstacle  to  the  further 
improvement  of  the  refracting  telescope  was  insuperable,  he 
turned  his  attention  to  a  form  of  telescope  which  had  been 
suggested  a  number  of  years  earlier,  and  in  which  the  image 
was  to  be  formed  by  reflection  from  a  concave  mirror  instead 
of  by  refraction.  Since  reflection  is  unattended  by  separa- 
tion into  elementary  colors,  it  is  evident  that  this  method 
would  eliminate  the  chief  defect  of  the  refractor.  With  his 
own  hands  Newton  constructed  a  small  telescope  on  this 
principle,  and  it  still  exists  in  the  possession  of  the  Royal 
Society.  This  little  instrument  seems  to  have  been  of  about 
the  same  power  as  Galileo's  telescope  with  which  he  dis- 
covered the  satellites  of  Jupiter,  but  it  was  hardly  more  than 
six  inches  in  length. 

Since  that  time  the  reflecting  telescope  has  had  a  remark- 
able history  of  development  in  the  hands  of  a  number  of  most 
skilful  mechanicians,  who  have  also  for  the  most  part  been 
distinguished  by  their  discoveries  in  physical  astronomy.  We 
may  therefore  advantageously  depart  from  the  chronological 
order  of  treatment,  and  follow  the  history  of  this  type  of 
instrument  to  the  present  time.  This  course  is  the  more 
natural  because  we  may  probably  regard  the  supremacy  of 


THE   TELESCOPE  59 

the  reflector,  undisputed  a  century  ago,  as  passed  forever. 
Only  in  the  province  of  astronomical  photography  does  it 
still  find  defenders. 

Even  after  Newton's  invention  was  made  public,  little  was 
done  toward  the  improvement  of  telescopes  for  half  a  cen- 
tury, until,  in  1723,  Hadley  presented  to  the  Royal  Society 
a  reflector  of  his  own  construction,  which  was  found  to  equal 
in  efficiency  the  Huyghens  refractor  of  123  feet  in  length. 
From  this  epoch  we  may  date  the  beginning  of  the  supremacy 
of  reflectors.  A  few  years  later  Short  commenced  his  career 
as  a  practical  optician,  and  for  thirty  years  he  remained 
unsurpassed  in  the  excellence  of  his  instruments.  During 
this  time  many  telescopes  more  powerful  than  the  best  of 
the  previous  century,  and  infinitely  more  convenient  to 
use,  had  been  made  and  scattered  throughout  Europe,  but 
there  was  also  a  singular  dearth  of  telescopic  discovery. 
Perhaps  men  thought  that  the  harvest  had  already  been 
gathered ;  or  perhaps  we  may  find  the  explanation  in  the  fact 
that  the  great  cost  of  telescopes  so  restricted  their  use  that 
the  impulse  to  discovery  by  their  means  was  confined  to  a 
very  small  class.  We  might  well  regard  the  latter  cause  as 
the  more  probable  one,  in  view  of  the  remarkable  manner 
in  which  the  standstill  in  this  branch  of  science  was  finally 
followed  by  a  brilliant  period  of  discovery  rivalled  alone  by 
that  of  Galileo. 

William  Herschel  was  born  in  1738,  in  Hanover.  In  1755 
he  left  his  native  country,  and,  going  to  England,  secured  a 
position  as  organist  in  Octagon  Chapel,  Bath,  where  we  find 
him  in  1766.  Here  he  became  so  profoundly  interested  in 
the  views  of  the  heavens  which  a  borrowed  telescope  of  mod- 
erate power  yielded  him,  that  he  tried  to  purchase  an  in- 
strument in  London.  The  cost  of  a  satisfactory  one  proving 
beyond  his  command,  he  determined  to  construct  a  telescope 
with  his  own  hands.  Thus  he  entered  upon  a  course  which 
was  to  reflect  honor  upon  himself,  his  country,  and  his  age, 
and  which  led  him  to  add  more  to  physical  astronomy  than 
any  other  one  man  has  added  before  or  since.  With  almost 


60  LIGHT 

inconceivable  industry  and  perseverance  he  cast,  ground, 
and  polished  more  than  four  hundred  mirrors  for  telescopes, 
varying  in  diameter  from  six  to  forty-eight  inches.  This  in 
itself  would  imply  a  busy  life  for  any  artisan ;  but  when  we 
remember  that  all  this  was  merely  subsidiary  to  his  main 
work  of  astronomical  discovery,  we  cannot  withhold  our 
admiration. 

Fortunately  for  science  as  well  as  for  himself,  he  made, 
early  in  his  career,  a  discovery  of  the  very  first  importance, 
which  attracted  the  attention  of  all  Christendom.  On  the 
night  of  March  13,  1781,  Herschel  was  examining  certain 
small  stars  in  the  constellation  of  Gemini,  with  one  of  his 
telescopes  of  a  little  more  than  six  inches  in  diameter,  when 
he  perceived  one  star  that  appeared  "  visibly  larger  than  the 
rest."  This  proved  to  be  a  new  planet  now  known  as  Uranus. 
In  the  following  year  the  discovery  led  to  his  appointment  as 
Astronomer  to  the  King,  George  III.,  with  a  salary  sufficient 
to  enable  him  to  devote  his  whole  time  to  astronomy. 

One  of  the  fruits  of  this  increased  leisure  was  the  con- 
struction of  an  instrument  far  more  powerful  than  had  been 
dreamed  of  by  his  predecessors,  namely,  a  telescope  four  feet 
in  diameter  and  forty  feet  in  length.  Commenced  in  1785, 
Herschel  dated  its  completion  as  August  28,  1789,  when  he 
discovered  by  its  means  a  sixth  satellite  of  Saturn,  and,  less 
than  a  month  later,  a  seventh  even  closer  to  the  planet  and 
smaller  than  the  sixth.  We  may  fairly  regard  this  achieve- 
ment as  marking  the  limit  of  progress  in  the  reflecting  tele- 
scope, for  although  at  least  one  as  large  is  now  in  use,  and 
one  even  half  as  large  again  has  been  constructed,  it  is 
more  than  doubtful  whether  they  were  ever  as  perfect  as 
Herschel's  at  its  best. 

There  has  been  one  improvement  in  the  reflecting  telescope 
since  the  time  of  Herschel  which  ought  not  to  be  left  un- 
noted here,  namely,  that  of  replacing  the  heavy  metal  mirror 
by  one  of  glass  rendered  even  more  highly  reflective  than 
the  old  mirrors  by  a  thin  coating  of  silver  deposited  by  chem- 
ical means  upon  the  polished  surface  of  the  glass.  The  great 


FIGURE   18. —  Herschel's  Telescope. 


62  LIGHT 

advantage  of  the  modern  form  of  reflector  lies,  not  so  much 
in  the  relative  rigidity  and  lightness  of  the  material,  as  in  the 
fact  that  the  surface  when  tarnished  can  be  readily  renewed 
by  the  simple  process  of  replacing  the  old  silver  film  by  a  new 
one;  whereas  in  the  metal  reflectors  a  tarnished  surface 
required  a  repetition  of  the  most  difficult  and  critical  portion 
of  the  whole  process  of  construction.  Moreover,  the  making 
of  such  telescopes  is  so  comparatively  simple  that  an  efficient 
reflector  is  far  less  expensive  than  is  a  refracting  telescope  of 
like  power,  so  that  this  type  may  be  regarded  as  particularly 
the  amateur's  instrument.  On  the  other  hand,  such  tele- 
scopes are,  like  their  predecessors,  extremely  inconstant, 
and  require  much  more  careful  attention  to  keep  them  in 
good  working  order.  It  is  for  such  reasons,  doubtless,  that 
silver-on-glass  reflectors  have  done  so  little  for  the  advance- 
ment of  astronomical  discovery.  In  astronomical  photog- 
raphy, however,  they  promise  to  do  much  on  account  of 
their  absolute  freedom  from  all  color  error  and  the  ease  with 
which  large  instruments  can  be  made.  In  England  some 
very  remarkable  photographs  of  nebulae  have  been  made  in 
recent  years  with  large  reflectors,  both  by  Mr.  Common  and 
by  Mr.  Roberts,  the  former  using  a  reflector  of  three  feet  in 
diameter  and  the  latter  one  of  twenty  inches.  Although  it 
is  impossible  to  assert  that  no  refracting  instrument  would 
yield  as  excellent  results,  it  remains  true  that  the  work  from 
these  sources  at  the  time  of  its  publication  proved  a  great 
surprise  to  all  astronomers. 

We  now  go  back  to  a  quarter  of  a  century  before  Herschel 
discovered  the  new  planet  —  to  the  very  year,  indeed,  when 
that  great  astronomer  first  set  foot  on  English  soil  —  in  order 
to  trace  the  history  of  another  form  of  telescope  which  has 
remained  unrivalled  for  the  last  half-century  in  the  more 
difficult  fields  of  astronomical  research,  and  which  to-day 
finds  its  most  perfect  development  in  the  instruments  at 
Williams  Bay,  at  Mt.  Hamilton,  at  Pulkowa,  at  Vienna,  and 
at  Washington. 

As  a  result  from  his  experiments,  Newton  had  declared 


THE   TELESCOPE  63 

that  separation  of  white  light  into  its  constituent  colors 
was  an  inevitable  accompaniment  of  deviation  by  refraction, 
and  consequently  the  shortening  of  the  unwieldy  refractor  of 
his  time  was  impracticable.  The  correctness  and  validity  of 
the  experiments  remained  unquestioned  for  nearly  a  century, 
but  finally  a  famous  German  mathematician,  Euler,  attempted 
to  throw  doubt  on  Newton's  conclusion  in  a  singular  way. 
His  argument  was  that,  since  the  eye  does  produce  colorless 
images  of  white  objects,  it  might  be  possible  by  the  proper 
selection  of  curves  so  to  combine  lenses  of  glass  and  of  water 
as  to  produce  a  telescope  free  from  the  color  defect.  Al- 
though Euler 's  premise  was  an  error,  since  the  eye  is  not  free 
from  dispersion,  his  efforts  had  the  effect  of  leading  to  a 
much  more  critical  study  of  the  phenomena  involved.  In 
this  John  Dolland,  an  English  optician,  met  with  brilliant 
success.  Repeating  an  experiment  of  Newton's  with  a  prism 
of  water  opposed  by  a  prism  of  glass,  he  found  that  deviation 
of  light  could  be  produced  without  dispersion  into  prismatic 
colors.  More  than  this,  he  found  that  the  two  varieties  of 
glass  then,  as  now,  common  in  England  —  namely,  crown  or 
common  window  glass,  and  flint  glass,  which  is  characterized 
by  the  presence  of  a  greater  or  less  quantity  of  tead  oxide  — 
possessed  very  different  powers  in  respect  to  dispersion. 
Thus,  of  two  different  prisms  of  these  two  varieties  of  glass 
which  would  deflect  the  light  by  the  same  angle,  that  made 
of  flint  glass  might  form  a  spectrum  quite  twice  as  long  as 
that  formed  by  the  other.  Hence,  if  a  prism  of  crown  glass 
deflecting  a  transmitted  beam  of  light  by  ten  degrees  were 
combined  with  one  of  flint  glass  which  would  deflect  the 
light  five  degrees  in  the  opposite  direction,  there  would 
remain  a  deflection  of  five  degrees  without  division  into 
colors.  It  also  follows  that  a  positive  lens  of  crown  glass 
combined  with  a  negative  lens  of  flint  glass  of  half  the  power 
would  yield  a  colorless  image.  Such  combinations  of  two  or 
more  substances  are  called  achromatic  systems.  Some  of  them 
are  illustrated  in  Figure  23,  page  97,  where  the  shaded  por- 
tions represent  the  flint  glass. 


64  LIGHT 

It  is  a  singular  fact,  which  is  worthy  of  note  at  this  point, 
that  more  than  twenty  years  before  Dolland's  success  achro- 
matic telescopes  had  been  invented  by  Mr.  Chester  Moor  Hall 
and  constructed  for  him;  but  for  some  reason  now  difficult 
to  explain,  the  possibility  of  such  construction  remained 
unknown  to  the  world  of  science  until  after  Dolland's  tele- 
scopes had  attained  fame. 

Why  the  greatest  philosopher  who  ever  lived,  and  at  the 
same  time  one  of  the  most  successful  experimenters,  should 
have  been  corrected  by  a  follower  employing  the  same  method 
of  investigation,  has  never,  to  my  knowledge,  been  explained. 
Still,  the  reason  does  not  seem  far  to  seek.  At  the  time  of 
Newton  the  two  substances  which  stood  at  the  extremes  of 
refractive  power,  and  which  at  the  same  time  were  so  abun- 
dant as  to  be  readily  procurable  for  investigation,  were  water 
and  comn^pn  glass.  There  were,  of  course,  a  vast  number  of 
gems  and  other  rare  substances  which  stand  quite  outside 
these  limits,  but  £hese  were  not  suited  for  the  methods  of 
research  at  command  at  that  time.  Now  it  so  happens  that 
these  two  substances,  water  and  common  glass,  do  have 
essentially  the  same  dispersive  power,  so  that  when  a  prism 
of  glass  is  cdfrected  by  a  prism  of  water  in  the  reverse  posi- 
tion, the  deviation  vanishes  with  the  color.  This  being  so, 
not  only  was  it  natural  to  conclude  that  the  indicated  de- 
duction was  a  general  law,  but  it  would  have  been  unphilo- 
sophical  to  doubt  it  without  further  experience.  It  is  quite 
conceivable  that  had  flint  glass  been  as  familiar  a  substance 
then  as  it  was  a  century  later,  the  true  relation  of  dispersion 
to  refractive  power,  or,  rather,  their  independence,  would 
not  have  escaped  the  earliest  and  most  acute  investigator  in 
this  field. 

For  a  long  time  this  ingenious  invention  of  Dolland  re- 
mained fruitless  for  astronomical  discovery,  although  it  was 
early  applied  to  use  in  meridian  instruments.  This  was 
due  to  difficulty  in  securing  sufficiently  large  and  perfect 
pieces  of  glass,  more  particularly  of  flint  glass,  to  meet  the 
demands  of  the  optician.  Not  until  after  the  beginning  of 


THE   TELESCOPE  65 

the  last  century  was  any  real  advance  exhibited  in  this  branch 
of  the  arts.  Even  then  success  appeared,  not  in  England 
or  France,  where  strenuous  efforts  had  been  made  to  improve 
the  quality  of  optical  glass,  but  in  Switzerland.  There  a 
humble  mechanic,  a  watchmaker  named  Guinand,  spent  many 
years  in  efforts,  long  unfruitful,  to  secure  a  method  of  mak- 
ing large  pieces  of  optically  useful  glass.  Just  what  degree 
of  success  he  attained  there  we  do  not  know,  though  from 
the  fact  that,  during  this  period  of  twenty  years  of  experi- 
menting, good  achromatic  telescopes  of  more  than  five  inches 
in  diameter  were  unknown,  we  must  conclude  that  his  suc- 
cess was  limited.  In  1805  he  joined  the  optical  establish- 
ment of  Fraunhofer  and  Utschneider  in  Munich.  Here  he 
remained  nine  years,  and  with  the  increased  means  at  his  dis- 
posal and  with  the  aid  of  Fraunhofer  he  perfected  his  methods 
so  far  that  the  production  of  large  disks  of  homogeneous  glass 
became  only  a  matter  of  time  and  cost.  It  is  not  too  much 
to  say  that  all  the  large  pieces  of  optical  glass  which  have 
since  been  produced  and  which  have  added  so  incalculably 
to  the  resources  of  the  scientific  investigator,  whether  made 
in  Germany,  France,  or  England,  owe  their  existence  to  the 
wonderful  patience  and  industry  of  this  Swiss  watchmaker, 
whose  trials  and  achievements  recall  those  of  Palissy. 

Fraunhofer  was  a  genius  of  high  order.  Although  he 
died  at  the  early  age^af  thirty -nine,  he  had  not  only  brought 
the  achromatic  telescope  to  ti  degree  of  optical  perfection 
which  made  it^rival  of  the  most  powerful  of  the  reflectors, 
and  so  far  inAoved  upon  the  method  of  mounting  it  that 
his  system  halPdisplaced  all  others,  but  he  also  made  some 
capital  discoveries  in  the  domain  of  physical  optics.  His 
great  achievement  as  an  optician  was  the  construction  of  an 
achromatic  telescope  nine  and  one-half  inches  in  diameter, 
with  which  the  elder  Struve  made  his  series  of  remarkable 
discoveries  and  measurements  of  double  stars  at  Dorpat. 
The  character  of  Struve's  work  demonstrates  the  excellence 
of  the  telescope,  and  shows  us  that  it  should  be  ranked  as  the 
equal  of  all  but  the  very  best  of  its  predecessors.  Indeed  it 


66  LIGHT 

may  be  fairly  concluded  that  not  more  than  one  or  two  tele- 
scopes of  greater  power,  and  those  made  and  used  by 
Herschel,  had  ever  existed,  while  in  convenience  for  use  the 
new  refractor  was  vastly  superior. 

For  a  long  time  Fraunhofer  and  his  successors,  Merz  and 
Mahler,  from  whom  the  second  equatorial  of  Pulkowa  and 
the  fifteen-inch  telescope  of  the  Harvard  Observatory  were 
procured,  remained  unrivalled  in  this  field  of  optics.  But 
they  have  been  followed  by  a  number  of  skilful  makers 
whose  products,  since  the  middle  of  the  century,  have  been 
scattered  throughout  the  world.  In  Germany,  Steinheil  and 
Schroeder;  in  France,  Cauchois,  Martini,  and  the  Henry 
brothers ;  in  England,  Cook  and  Grub ;  and  in  this  country, 
the  Clarks  and  Brashear  —  each  has  produced  one  or  more 
great  telescopes  which  have  rendered  his  name  familiar  to 
all  readers  of  astronomical  literature.  Of  these  the  Clarks, 
father  and  son,  have  beyond  a  doubt  won  the  first  place, 
whether  determined  by  the  character  of  the  discoveries  made 
with  their  instruments  or  by  the  fact  that  the  two  most 
powerful  telescopes  in  existence  —  the  refractor  of  three-feet 
aperture  of  the  Lick  Observatory  in  California,  and  the 
forty-inch  telescope  of  the  Yerkes  Observatory  in  Wisconsin 
—  were  made  by  them.  The  most  notable  discoveries  made 
with  their  telescopes  are  the  satellites  of  Mars,  the  fifth 
satellite  of  Jupiter,  and  the  companion  star  to  Sirius;  but 
besides  these  there  is  a  long  list  of  double  stars  of  the  most 
difficult  character  discovered  by  the  makers  themselves,  by 
Dawes  in  England,  and,  far  the  most  notable  of  all,  by 
Burnham  in  our  own  country. 

The  Yerkes  telescope,  which  now  stands  as  the  highest 
representative  of  an  art  itself  the  growth  of  nearly  three 
centuries  of  human  endeavor,  is  represented  in  the  accom- 
panying picture.  It  is  vastly  more  complicated  than  its 
famous  Fraunhofer  prototype,  but  it  does  not  differ  in  any 
fundamental  principle.  Like  that,  it  has  an  axis  parallel  to 
the  axis  of  rotation  of  the  earth,  to  the  upper  end  of  which 
there  is  attached  another  axis  at  right  angles  to  it;  the  latter 


FIGURE  19.  —  Fraunhofer's  Equatorial  Telescope. 


68  LIGHT 

carries  at  one  end  the  telescope  tube,  which  is  supported  not 
far  from  its  middle  point.  If  the  instrument  is  rotated  on 
the  first  described  axis,  which  is  called  the  polar  axis,  its  line 
of  sight  will  describe  a  circle  on  the  heavens  parallel  to  the 
equator;  it  is  this  property  which  gives  the  name  equatorial 
to  such  a  mounting.  Since,  in  their  diurnal  motion,  the 
heavenly  bodies  appear  to  describe  circles  parallel  to  the  equa- 
tor, to  follow  a  star  once  found  in  the  telescope  it  is  only 
necessary  to  rotate  the  instrument  on  the  polar  axis  alone  and 
at  the  same  rate  as  the  apparent  motion  of  the  star.  This 
motion  can  be  secured  with  wonderful  regularity  and  smooth- 
ness by  a  small  motor  —  shown  in  part  within  the  hollow 
cast-iron  pier  —  called  a  driving  clock.  To  each  axis  is 
attached  a  large  divided  circle  which  enables  an  astronomer 
to  find  any  object  in  the  heavens  whose  place  is  known,  or  to 
determine  the  position  of  a  previously  unknown  object  when 
brought  to  the  centre  of  the  field  of  the  telescope.  Impor- 
tant accessories  to  a  large  instrument  are  one  or  more  small 
telescopes  attached  to  the  eye  end,  with  their  lines  of  sight 
parallel  to  that  of  the  great  telescope.  These  enable  the 
astronomer  to  find  an  object  and  point  his  large  instrument 
with  greater  facility,  and  are  appropriately  called  finders. 

A  comparison  of  Figures  17  to  20,  inclusive,  is  highly 
instructive.  In  the  first  we  have  the  telescope  of  the  seven- 
teenth century  in  the  state  of  highest  development.  Her- 
schel's  great  telescope  was  not  only  the  most  powerful  of  the 
following  century,  but  it  was  also  perfectly  typical  of  the 
best  that  that  century  produced.  Finally,  the  third  and 
fourth  of  these  figures  represent,  respectively,  the  earliest  and 
the  most  recent  of  modern  powerful  refractors,  separated 
in  time  of  construction  by  almost  exactly  three-fourths  of  a 
century. 

There  are  two  points  in  the  theory  of  the  telescope  which 
are  of  fundamental  importance,  and  which,  by  means  of  the 
theoretical  considerations  developed  in  the  preceding  chap- 
ters, can  be  readily  understood.  The  first  determines  the 
range  of  magnifying  powers  for  any  given  instrument,  and 


,      r^  ,,         V 


FIGURE  20.  —  Equatorial  Telescope  of  the  Yerkes  Observatory. 


70  LIGHT 

the  second  enables  us  to  find  the  absolute  efficiency  of  a 
telescope. 

In  Chapter  II.  the  essentials  of  a  telescope  were  shown  to 
be  an  object-glass  of  low  power  for  forming  an  image,  and 
an  eyepiece,  or  ocular,  for  magnifying  this  image.  But  this 
ocular  will  form  a  real  image  of  everything  in  front  of  it 
which  is  sufficiently  remote;  it  will  consequently  form  an 
image  of  the  object-glass.  Such  an  image  can  be  always 
seen  close  to  the  ocular  if  the  telescope  is  directed  toward 
a  bright  surface,  as,  for  example,  toward  the  sky.  Now  a 
simple  calculation  shows  that  the  ratio  of  the  diameter  of  the 
objective  to  that  of  this  ocular  image  is  exactly  equal  to  the 
magnifying  power  of  the  telescope.  This  is  by  far  the  best 
method  of  determining  the  magnification.  This  ocular  circle 
is  also  the  smallest  area  through  which  the  light  passes  after 
leaving  the  telescope,  and  thus  marks  the  best  position  of  the 
eye  of  the  observer.  If  the  pupil  of  the  eye  is  smaller  than 
the  ocular  circle,  it  is  obvious  that  not  all  the  light  which  is 
transmitted  through  the  objective  will  reach  the  retina.  The 
lowest  power,  therefore,  which  can  be  used  with  full  advan- 
tage on  any  telescope  is  found  by  multiplying  the  diameter 
of  the  objective  in  inches  by  five  or  six.  Thus  the  lowest 
power  which  can  be  used  with  the  Lick  telescope  and  have  it 
retain  its  superiority  over  smaller  instruments  is  from  180 
to  220. 

On  the  other  hand,  experience  teaches  that  vision  begins 
to  be  sensibly  impaired  when  the  ocular  circle  is  much  less 
than  one-thirtieth  of  an  inch  in  diameter;  consequently  the 
highest  power  which  is  generally  available  is  found  by  mul- 
tiplying the  diameter  of  the  objective  in  inches  by  thirty  or 
forty.  For  example,  the  magnifying  powers  employed  on 
the  twenty -six  inch  equatorial  at  Washington  are  practically 
confined  to  400,  600,  and  900,  making  the  diameters  of  the 
corresponding  ocular  circles  1/15,  1/23,  and  1/35  of  an  inch. 
It  was  with  the  lowest  of  these  powers  that  both  the  satellites 
of  Mars  were  discovered  by  Professor  Hall. 

Thus  we   see   that    the  range  of  useful   powers    in    any 


THE   TELESCOPE  71 

telescope  is  practically  confined  to  from  five  to  forty  times  the 
number  expressing  the  diameter  of  the  objective  in  inches. 
The  lowest  limit  has  been  previously  established  by  theoretical 
considerations  and  does  not  admit  of  question ;  but,  as  far  as 
appears  in  what  precedes,  the  upper  limit  is  quite  empirical, 
and  naturally  suggests  the  question  as  to  whether  we  may 
not  in  the  future  so  perfect  the  telescope  that  smaller  instru- 
ments can  bear  higher  powers  and  thus  do  the  work  of 
greater  ones.  The  answer  to  this  eminently  practical  ques- 
tion is  an  unqualified  negative.  The  following  reasoning 
demonstrates  that  the  upper  limit  of  magnification  is  fixed  by 
the  nature  of  light  itself,  and  increase  of  power  necessarily 
involves  an  increase  of  size  in  the  telescope. 

It  is  obvious  that  a  telescope  directed  to  a  star  gives  pre- 
cisely the  condition  represented  in  Figure  14,  page  32,  that 
is,  a  concave  wave-front  with  its  centre  at  the  place 
which  marks  the  image  of  the  star.  The  diameter  of  the 
wave-front  just  after  passing  the  objective,  represented  by 
the  distance  ab  in  that  diagram,  is  the  same  as  that  of  the 
objective.  Hence,  it  follows  that  the  image  of  the  star  must 
consist  of  a  disk  surrounded  by  a  series  of  concentric  rings 
precisely  similar  in  appearance  to  the  artificial  star  seen 
through  a  needle  hole  in  a  card.  In  the  case  of  a  slit  of 
width  ab  it  was  there  shown  that  the  distance  from  the  centre 
of  the  image  to  the  nearest  dark  space  is  equal  to  ap^  the 
radius  of  the  wave-front,  multiplied  by  the  length  of  a  wave 
of  light  and  divided  by  ab.  The  more  intricate  analysis  for 
a  circular  aperture  shows  that  for  this  case  the  term  should 
be  multiplied  by  1.2.  Thus  the  image  of  a  star  in  a  tele- 
scope consists  of  a  disk  whose  diameter  is  2.4  F\/D,  where 
F  is  the  focal  length  and  D  the  diameter  of  the  objective. 
The  angular  diameter  of  the  image  is  this  quantity  divided 
by  F,  which  for  a  one -inch  telescope,  assuming  that  the 
length  of  a  wave  of  light  is  1/46000  of  an  inch,  would  be 
10".  75;  hence  two  stars  separated  by  an  interval  of  10".  75 
would  just  appear  to  touch,  if  the  light  could  be  traced  quite 
to  the  place  of  the  dark  ring.  In  fact,  however,  the  edge 


72  LIGHT 

of  such  a  disk  is  so  faint  that  it  appears  much  smaller  than 
indicated  by  this  reasoning,  and,  under  the  most  favorable 
circumstances,  at  somewhat  less  than  half  this  distance  two 
stars  can  be  just  seen  as  two  distinct  objects.  Conse- 
quently, to  find  the  closest  double  stars  which  can  be  seen 
with  any  perfect  telescope  we  divide  4  ".56 l  by  the  diameter 
of  the  objective  expressed  in  inches.  This  rule  is  found  to 
be  quite  accurate  for  even  the  largest  telescopes,  if  perfect; 
for  example,  under  favorable  circumstances,  the  Washington 
telescope  of  twenty-six  inches  diameter  ought,  according  to 
the  rule,  to  divide  stars  separated  by  an  angular  interval  of 
0".20  only,  and  in  fact  double  stars  0".23  apart  have  been 
seen  as  separated. 

It  is  clear  that  if  the  images  of  two  points  are  not  sepa- 
rated they  cannot  be  made  to  appear  so  by  mere  increase 
of  magnification  by  the  ocular,  which  is  quite  analogous  to 
the  familiar  fact  that  beyond  a  very  limited  extent  one  can 
see  nothing  more  in  a  photograph  by  increasing  the  magnifi- 
cation under  which  it  is  examined.  The  angular  separation 
is  thus  shown  to  depend  on  the  diameter  of  the  objective 
alone,  and  it  is  this  element,  therefore,  which  determines 
the  upper  limit  of  the  power  of  a  telescope  as  well  as  the 
lower  limit. 

The  preceding  pages  make  quite  evident  that  the  present 
degree  of  excellence  in  the  telescope  has  been  attained  only 
after  prolonged  efforts,  extending  through  a  period  of  many 
generations  and  participated  in  by  a  host  of  inventors,  among 
whom  are  a  number  who  have  been  illustrious  in  other 
fields  of  work.  It  will  not  be  without  interest  to  consider 
the  reasons  why  this  progress  has  been  so  slow  in  appear- 
ance; why  even  now  the  greater  number  of  telescopes  in 
existence  are  distinctly  bad  when  judged  by  a  high  standard ; 
how  it  was  that  for  a  time,  very  brief  it  is  true,  the  glass- 
maker  was  in  advance  of  the  demands  of  the  optician ;  and, 
finally,  why  the  first  of  the  great  modern  objectives  was  in 

1  This  is  the  value  fixed  upon  by  the  astronomer  Dawes  after  loiig  experience 
with  excellent  telescopes. 


THE   TELESCOPE  73 

the  hands  of  the  most  skilful  telescope-maker  of  Great 
Britain  for  seven  years  without  satisfying  that  optician  as  to 
its  ultimate  excellence.  These  questions  cannot  be  answered 
in  a  word,  nor  do  they  admit  of  complete  answers ;  but  much 
may  be  gained  by  recognizing  that  they  involve  two  distinct 
kinds  of  reasons,  namely,  those  depending  upon  purely  tech- 
nical difficulties  of  construction,  and  difficulties  which,  since 
they  depend  upon  the  solution  of  a  definite  mathematical 
problem,  may  be  called  theoretical  difficulties.  We  shall 
consider  these  in  turn. 

The  art  of  lens  making  can  certainly  be  traced  back  to  the 
thirteenth  century,  although  the  methods  employed  even  as 
late  as  the  beginning  of  the  seventeenth  were  so  crude  that 
Galileo  found  the  utmost  difficulty  in  making  a  lens  suffi- 
ciently good  to  bear  a  magnifying  power  of  thirty  times. 
We  may  be  quite  sure  that  he  could  command  the  aid  of  the 
best  artisans  in  Europe  and  a  complete  knowledge  of  their 
methods ;  still  he  was  obliged  to  train  his  own  hands  to  attain 
a  necessary  and  higher  degree  of  skill  than  was  possessed 
by  any  of  his  contemporaries.  A  most  illuminating  fact 
with  regard  to  the  relative  state  of  the  industrial  arts  in  those 
times  as  well  as  in  the  present  is  that  for  a  few  cents  it  is  now 
easy  to  secure  a  pair  of  spectacle  lenses  which  would  form  a 
telescope  rivalling  if  not  surpassing  in  power  that  first  and 
most  famous  of  all  telescopes.  Not  until  the  lapse  of  another 
generation  was  there  such  improvement  in  the  technique  of 
lens  making  that  further  advances  in  astronomical  discovery 
became  possible.  This  slow  progress  is  to  be  explained  by 
the  extremely  critical  requirements  for  a  good  lens.  A 
departure,  by  a  fraction  of  a  hundred-thousandth  of  an  inch, 
of  a  material  portion  of  the  surface  from  the  correct  geomet- 
rical form  will  greatly  impair  the  performance  of  an  objec- 
tive. But  even  at  the  present  day  the  limit  of  accurate 
measurement  may  be  set  at  about  a  one  hundred-thousandth 
part  of  an  inch,  while  it  is  quite  probable  that  ten  times  that 
amount  was  vanishingly  small  to  the  artisan  of  a  century  and 
more  ago.  It  was  necessary,  therefore,  to  devise  a  method 


74  LIGHT 

of  polishing  —  for  it  was  comparatively  easy  to  grind  a  sur- 
face accurately  —  which  should  keep  the  surface  true  within 
a  limit  far  transcending  the  range  of  possible  measurements. 
Huyghens  seems  first  to  have  succeeded  in  this  by  polishing 
upon  a  paste  which  was  formed  to  the  glass  and  then  dried, 
and  by  employing  only  a  portion  of  the  centre  of  a  large  lens. 
In  Italy,  at  about  the  same  time,  Campani  developed  a  system 
that  he  most  jealously  guarded  as  a  secret  up  to  his  death, 
after  which  it  became  known  from  an  inspection  of  the  tools 
and  unfinished  work  in  his  shop.  The  peculiarity  of  his 
method  consisted  in  polishing  his  lenses  with  a  dry  powder 
on  paper  cemented  to  the  tool  upon  which  the  lens  had  been 
ground.  This  practice  still  survives  in  Paris  to  the  exclu- 
sion of  almost  all  others,  and  it  is  in  many  respects  the  best 
for  work  which  does  not  demand  the  highest  scientific  accu- 
racy. This  qualification  does  not  mean  to  imply  that  the 
Parisian  opticians  have  failed  to  produce  work  of  a  high 
degree  of  excellence,  but  only  that  the  method  is  not  free 
from  serious  mechanical  objections,  and  that  nowhere  else 
has  it  been  employed  by  opticians  who  have  attained  distinc- 
tion as  telescope-makers. 

Newton  seems  to  have  been  the  first  one  to  make  use  of  a 
method  which  has  since  been  developed  to  a  state  of  surpris- 
ing delicacy.  Casting  about  to  find  a  means  which  should 
be  sufficiently  "tender,"  to  borrow  his  own  expression,  for 
polishing  the  soft  speculum  metal  of  his  reflecting  telescope, 
he  fixed  upon  pitch,  shaped  to  the  mirror  while  warm,  as  a 
bed  to  hold  the  polishing  powder.  But  the  enormous  value 
of  this  and  similar  substances  lies  not  so  much  in  the  com- 
parative immunity  which  it  gives  from  scratching,  but  to  the 
fact  that  under  slowly  changing  forces  it  behaves  like  a 
liquid,  while  under  those  of  short  duration  it  acts  as  though 
it  were  a  hard  and  brittle  solid.  Thus  it  is  possible  with 
such  a  polisher  slowly  to  alter  the  shape  of  a  lens  while  on 
the  tool,  and  to  correct  errors  which  are  found  by  inspection 
of  the  optical  image  where  they  may  be  very  sensible,  al- 
though due  to  insensible  departures  of  the  lens  surface  from 


THE   TELESCOPE  75 

the  proper  form.  In  view  of  the  thinness  of  the  layer  which 
he  used,  it  is  perhaps  doubtful  whether  Newton  recognized 
this  particular  property  of  his  pitch  polisher,  and  it  is 
also'  a  question  whether  he  contemplated  the  refinement  of 
shaping  his  mirror  to  the  form  of  a  paraboloid  of  revolution, 
which  was  clearly  indicated  as  the  proper  form  and  which 
necessarily  demanded  this  property.  However  this  may  be, 
there  is  no  doubt  that  the  merit  of  perfecting  the  process 
and  bringing  it  to  its  present  condition  belongs  to  the  Eng- 
lish of  the  eighteenth  century  and  the  early  part  of  the  last 
century.  In  the  Philosophical  Transactions  of  the  Royal 
Society  we  find  many  long  papers  relating  to  this  art,  con- 
tributed by  skilful  and  successful  amateurs,  and  we  may 
regard  the  technique  of  the  art  of  lens  making  as  essentially 
complete  by  the  middle  of  the  century  just  ended  and  as 
common  property,  so  that  success  no  longer  depends  upon 
the  possession  of  some  secret  of  manipulation. 

In  this  hasty  review  of  the  development  of  the  telescope, 
no  one  interested  in  the  history  of  scientific  progress  can 
fail  to  be  struck  with  the  prominence  of  the  three  greatest 
philosophers  of  the  seventeenth  century.  That  each  one  of 
these  men  was  content  to  employ  his  own  hands  in  advancing 
a  purely  mechanical  art,  adds  to  our  admiration  rather  than 
detracts  from  it.  The  interest  is  heightened  by  the  knowl- 
edge that  specimens  of  the  handiwork  of  all  three  still 
survive,  and  are  treasured  possessions  of  various  scientific 
societies.  Who  would  not  be  eager  to  add  to  such  treasures 
a  lens  made  by  Spinoza?  It  is  hardly  probable  that  this 
philosopher  advanced  an  art  which  enabled  him  to  secure  a 
modest  living  while  engaged  upon  those  philosophical  specu- 
lations which  will  remain  a  monument  to  philosophy  as  long 
as  interest  in  intellectual  affairs  survives,  but  such  a  memento 
might  possibly  command  even  a  more  wide-spread  interest  than 
any  of  those  named.  We  may  also  recall  here,  quite  appro- 
priately, the  name  of  another  great  philosopher,  a  contem- 
porary of  both  Newton  and  Spinoza,  Descartes,  who  first 
published  the  true  law  of  refraction,  and  proposed  methods  of 


76  LIGHT 

shaping  lenses  to  forms  which  his  high  mathematical  powers 
led  him  to  think  superior  to  the  spherical  form;  but  as  his 
contributions  have  not  proved  of  real  value  in  the  development 
of  the  art,  we  may  be  content  with  this  brief  mention. 

The  second  type  of  difficulties,  namely,  those  which  have 
been  styled  theoretical,  we  can  treat,  without  the  language 
of  mathematics,  only  in  a  very  general  manner;  but  even  with 
this  limitation  we  may  hope  to  derive  some  idea  of  what  the 
problems  have  been  and,  still  more  important  from  our  stand- 
point, what  we  may  hope  for  in  future  improvements. 

The  obvious  requirement  in  a  perfect  objective  is  that  light 
coming  from  a  point  in  the  object  should  be  concentrated  at 
a  point  in  the  image ;  but  this  requirement  combined  with  a 
prescribed  focal  length  may  be  defined  by  three  conditions, 
namely,  freedom  from  color  error,  absence  of  spherical  aber- 
ration, and  assigned  optical  power.  Now  consider  what 
provisions  are  at  command  for  satisfying  these  conditions. 
They  are  four  surfaces  which  must  be  very  nearly  spherical 
but  may  have  any  radii  we  please,  the  two  thicknesses,  and 
the  single  distance  separating  the  two  lenses.  These  number 
seven,  and  exhaust  the  list  for  a  binary  system.  As  a  matter 
of  fact,  however,  on  account  of  the  cost  of  the  material  and 
the  want  of  perfect  transparency  in  glass,  it  is  necessary  to 
make  large  lenses  as  thin  as  possible ;  and  even  the  element 
of  separation  of  the  two  lenses  is  found  to  be  unavailable  in 
practice,  for  reasons  which  will  be  touched  upon  later;  at 
least  this  statement  is  true  for  two-lens  systems,  which  engage 
our  attention  at  this  moment.  Thus  there  are  left  only  the 
four  radii  which  may  be  regarded  as  constants  susceptible  of 
arbitrary  choice.  But  these  are  more  than  enough  to  meet 
three  conditions,  whence  our  problem,  in  its  given  form  is 
indeterminate,  and  it  becomes  necessary  to  add  another  con- 
dition for  the  sake  of  definiteness.  Of  course  this  new  condi- 
tion will  be  chosen  so  as  to  add  some  other  desirable  property 
to  the  objective,  and  it  is  in  the  question  as  to  what  addi- 
tional property  is  most  desirable  that  we  find  room  for  wide 
diversity  of  opinion.  Clairault  proposed  to  make  the  fourth 


THE   TELESCOPE  77 

condition  that  the  two  adjacent  surfaces  should  fit  together 
and  thus  admit  of  cementing  with  a  transparent  cement.  This 
construction  is  of  great  value  in  small  objectives  and  is  very 
largely  used,  but  in  large  telescopes  it  is  found  impossible 
to  cement  the  components  of  the  objective  without  changing 
their  shapes  to  such  an  extent  as  quite  to  spoil  their  perform- 
ance. At  a  period  still  early  in  the  history  of  the  modern 
telescope  (1821),  Sir  John  Herschel  published  a  very  impor- 
tant paper,  in  which  he  made  the  fourth  condition  that  the 
spherical  aberration  should  vanish  for  objects  at  a  very  great 
distance  and  also  for  those  at  a  moderate  distance.  In  this 
paper  he  computed  a  table,  afterward  greatly  extended  by 
Professor  Baden  Powell,  for  the  avowed  purpose  of  aiding 
the  practical  optician.  It  was  doubtless  this  feature  that  for 
a  considerable  time  brought  his  construction,  which  is  cer- 
tainly not  recommended  by  the  formally  stated  condition,  into 
more  general  use  than  any  other.  But  as  all  these  tables  were 
derived  from  calculations  which  wholly  disregarded  the  thick- 
ness and  separation  of  the  lenses,  they  could  yield  a  rather 
rough  approximation  only,  and  it  is  quite  possible  that  the 
discredit  with  which  opticians  have  received  the  dicta  of 
mathematicians  concerning  their  instruments  may  have  been 
in  part  due  to  this  very  fact.  Fraunhofer,  whose  remarkable 
achievements  in  this  field  have  already  been  touched  upon,  is 
said  to  have  imposed  the  condition  that  the  magnifications  for 
all  zones  of  the  objective  should  be  alike,  and  thus  made  the 
problem  determinate ;  he  does  not  seem  to  have  anywhere  pub- 
lished his  method  or  results.  This  condition  is  of  high  im- 
portance where  a  considerable  field  of  view  is  required,  as 
in  photographic  telescopes,  those  used  for  transit  instruments, 
and,  in  general,  those  designed  for  the  better  class  of  survey- 
ors' instruments.  It  is  a  singular  fact  that  in  its  ultimate 
solution  this  very  philosophical  condition  corresponds  to  a 
second  approximation  with  that  of  Herschel,  which  hardly 
pretended  to  be  more  than  a  convenient  way  of  avoiding  a 
difficulty  in  the  problem  due  to  indefiniteness ;  so  true  is 
this  that  no  one  could  tell  from  inspection  of  a  finished 


78  LIGHT 

objective  whether  one  or  the  other  condition  were  the  guid- 
ing principle  in  the  construction. 

A  particularly  interesting  proposal  for  the  necessary  fourth 
condition  is  that  of  Gauss ;  but  to  describe  it  will  demand  an 
immediate  statement  of  a  fact  which  is  to  receive  attention 
a  little  later.  We  have  already  tacitly  assumed  that  freedom 
from  color  error  is  always  attainable;  but  this  is  so  far  from 
true  that  as  a  matter  of  fact  only  a  very  moderate  approxima- 
tion to  this  ideal  state  can  be  reached.  A  general  statement 
of  this  kind  may  be  made :  In  any  achromatic  objective  the 
focal  point  of  a  given  wavelength  is  found  to  agree  with  that 
of  some  other  one  wavelength,  but  with  one  only.  Now  the 
Gaussian  condition  was  that  the  spherical  aberration  for  two 
such  different  wavelengths  should  vanish  simultaneously. 
This  seems  to  have  been  a  tour  de  force  of  a  mathematician, 
not  a  sober  suggestion  of  an  improvement  in  construction, 
for  in  point  of  fact  the  solution  yields  a  form  which  is  far 
from  being  a  good  one.  It  was  generally  believed  that  this 
condition  could  not  be  fulfilled ;  therefore  Gauss,  who  seemed 
particularly  fond  of  doing  what  all  the  rest  of  the  world 
deemed  impossible,  straightway  did  it. 

Such  are  the  more  famous  proposals  for  giving  definiteness 
to  the  foregoing  problem ;  but  they  do  not  exhaust  the  list. 
Others  have  been  suggested  by  the  astronomers  Litrow  and 
Bohnenberger,  but  a  detailed  description  of  their  solutions 
would  add  little  of  essential  interest.  Considering  that  none 
of  the  solutions  have  predominating  merit,  it  is  hardly  sur- 
prising that  the  practical  optician  has  followed  the  line  of 
least  resistance  and  adopted  a  form  which  costs  him  less 
labor  than  the  others  and  is  sensibly  their  equal.  By  mak- 
ing the  crown  lens  equiconvex  the  trouble  of  making  one 
pair  of  tools  is  saved,  which  probably  is  quite  sufficient  to 
explain  the  vogue  of  that  construction.  Of  course  this 
reason  should  have  no  weight  with  the  astronomer  who 
seeks  the  best  instrument  it  is  possible  to  make ;  therefore, 
as  there  seems  to  be  nothing  else  to  recommend  the  construc- 
tion, it  will  probably  be  used  little  in  the  future. 


THE   TELESCOPE  79 

The  reason  for  so  much  futile  work  on  the  theory  of  the 
telescope  objective  is  not  far  to  seek.  It  had  always  been 
tacitly  assumed  that  the  condition  of  color  correction,  one  of 
those  which  serves  to  define  the  values  of  the  arbitrary  con- 
stants, was  readily  determinable  —  in  fact,  one  of  the  data 
of  the  problem  —  whereas  it  is  the  finding  of  just  this  datum 
which  has  offered  peculiar  difficulties.  Fraunhofer  brought 
all  the  resources  at  the  command  of  his  exceptional  genius 
to  bear  upon  this  point  and  frankly  failed,  although  in  the 
effort  he  made  a  splendid  discovery  which  has  assured  a  per- 
manence to  his  fame  no  less  enduring  than  the  history  of 
science  itself  —  the  discovery  of  the  dark,  or  Fraunhofer, 
lines  in  solar  and  stellar  spectra.  Gauss  proposed  the  con- 
dition that  the  best  objective  must  be  that  which  produces 
the  most  perfect  concentration  of  light  about  the  place  of  the 
geometrical  image  of  a  point,  just  as  the  best  rifle  practice 
is  that  which  produces  the  maximum  concentration  of  hits 
about  the  centre  of  the  target.  That  this  principle  is  inade- 
quate appears  at  once  from  the  consideration  that  even  if  we 
take  as  much  as  a  tenth  of  all  the  light  from  ah  object  and 
divert  it  wholly  from  the  field  of  vision,  the  telescope  may 
still  be  practically  perfect.  All  of  Herschel's  telescopes  did 
much  worse  than  this.  But  if  we  take  the  same  proportion  of 
light  and  concentrate  it  in  the  immediate  vicinity  of  the  image, 
the  telescope  will  prove  worthless.  The  true  principle,  briefly 
stated,  is  that  the  weighted  mean  position  of  the  light  should 
correspond  with  the  place  of  the  geometrical  image,  the 
weights  being  assigned  according  to  the  relative  importance 
of  the  different  wavelengths  for  the  particular  end  in  view. 
For  example,  if  it  is  desired  to  affect  the  retina  it  is  necessary 
to  recognize  that  the  yellow  waves  are  far  more  effective 
than  those  of  other  colors,  and  for  purposes  of  photography 
this  pre-eminence  is  transferred  to  the  blue  or  violet  waves. 

Perhaps  the  true  difficulty  with  most  of  the  theoretical 
discussions  of  this  important  problem  is  this:  There  is  in 
them  no  recognition  of  the  relative  weight  or  importance  of 
unavoidable  errors.  At  the  very  outset  of  his  task  the  opti- 


80  LIGHT 

cian  is  confronted  by  the  fact  that  absolute  elimination  of 
color  error  in  a  binary  system  is  impossible,  because  no  two 
substances  are  known  which  have  dispersive  ratios  indepen- 
dent of  the  color.  He  can,  it  is  true,  reduce  the  color  error 
of  the  old  single-lens  type  of  telescope  hundreds  of  times, 
and  hence  the  length  of  the  telescope  tens  of  times ;  but  the 
fact  that  he  must  stop  at  a  point  far  short  of  perfection  puts 
an  end  to  still  further  shortening,  and  in  considerable  tele- 
scopes leaves  the  minimum  ratio  of  length  to  diameter  not  far 
from  15  to  1.  This  restriction  being  recognized,  we  may 
revise  our  limiting  conditions.  They  now  become,  first, 
fixed  focal  length;  second,  best  attainable  color  correction; 
third,  freedom  from  spherical  aberration  for  a  particular 
wavelength  of  light.  This  still  leaves,  as  before,  one  of  the 
necessary  four  lacking.  What  should  determine  the  choice 
of  the  fourth  condition?  Surely  there  is  only  one  rational 
guide.  Consider  the  residual  errors  and,  with  proper  regard 
to  their  relative  importance,  make  the  final  condition  such 
as  to  reduce  these  errors  to  the  smallest  possible  values. 
But  the  only  remaining  errors  for  an  image  in  the  axis  are 
secondary  color  error  and  spherical  aberration  for  colors  other 
than  that  for  which  it  is  eliminated,  or,  more  scientifically 
stated,  chromatic  difference  of  spherical  aberration.  As  to 
which  of  these  is  the  more  grave  defect  depends  upon  the  use 
to  which  the  objective  is  to  be  put.  If  it  is  a  high-power 
microscope  objective,  it  is  certainly  the  second.  If  it  is  an 
objective  to  be  used  for  photographing  at  a  considerable  angu- 
lar distance  from  the  axis,  the  question  loses  its  physical  sig- 
nificance, since  the  consideration  of  eccentric  refraction  is 
excluded.  But  if  the  objective  is  to  be  for  a  visual  telescope, 
there  is  no  question  that  the  defect  of  secondary  color  error 
is  indefinitely  more  serious.  The  fourth  and  determining 
condition  must  therefore  be  improvement  of  color  correction 
to  the  last  attainable  limit.  With  this  end  in  view  the 
improvement  may  be  made  in  part  by  a  careful  selection  of 
material,  for  the  available  glasses  are  by  no  means  of  unvarying 
imperfection  in  this  particular. 


THE    TELESCOPE  81 

The  question  as  to  what  further  improvement  may  be  hoped 
for  in  the  future  of  this  most  important  instrument  of  re- 
search is  a  natural  and  interesting  one.  The  absolute  power 
has  been  shown  to  depend  upon  the  diameter  of  the  objective 
alone,  if  errors  of  construction  are  perfectly  eliminated; 
hence  the  question  leads  at  once  to  the  consideration  of  the 
probability  of  securing  useful  disks  of  glass  much  larger 
than  those  hitherto  employed,  for  we  may  with  little  doubt 
assert  that  the  ablest  opticians  are  competent  to  shape  lenses 
notably  larger  than  any  now  in  existence,  to  the  necessary 
degree  of  precision.  It  is  obvious  that  no  limit  can  be  set 
to  the  possible  achievements  of  glass-makers  in  the  future ; 
but  we  can  state  positively  that  were  there  no  limit  to  the 
size  of  sufficiently  homogeneous  glass  at  command,  we  should 
finally  reach  dimensions  of  which  the  necessary  weight 
would  produce  such  distortions  that  further  increase  would 
carry  with  it  no  increase  in  optical  efficiency.  Whether  we 
have  begun  to  approach  this  limit  imposed  by  the  finite  rigid- 
ity of  glass  is  a  question  which  cannot  be  answered  with 
positiveness,  but  some  observations  with  the  largest  refractor 
now  in  existence  may  be  interpreted  to  point  in  this  direction. 

There  remains  to  consider  what  an  increase  in  perfection  of 
telescopes  of  ordinary  power  can  yield.  If  it  were  possible 
to  secure  two  varieties  of  glass  which  should  possess  this 
highly  desired  independence  of  dispersive  ratio  and  color,  we 
might  eliminate  the  most  serious  error  remaining  in  the  tele- 
scope, provided  always  that  the  other  physical  properties 
of  the  materials  were  suitable.  The  highly  scientific  glass- 
makers  of  Jena  have  carried  on  experiments  to  this  end  for 
many  years,  but,  it  must  be  confessed,  with  rather  discour- 
aging results.  It  is  true  that  several  pairs  of  glasses  have 
been  made  which  materially  reduce  the  defect,  but  either  one 
of  the  glasses  has  proved  to  be  perishable  or  too  hygroscopic 
for  use,  or,  in  the  most  promising  case,  the  refractive  powers 
differ  so  little  that  the  chromatic  difference  of  spherical  aber- 
ration for  telescopes  of  moderate  length  constitutes  a  defect 
worse  than  that  corrected.  The  employment  of  three  kinds 


82  LIGHT 

of  glass,  although  it  admits  of  a  fairly  complete  solution  of 
the  problem,  does  not  seem  to  promise  to  be  useful  in  large 
telescopes,  on  account  of  the  much  greater  length  enforced 
and  the  increased  inconvenience  of  reflections  from  the  greater 
number  of  surfaces.  The  possibility  of  accomplishing  the 
same  end  in  other  ways  must  be  left  to  another  place  for 
discussion.  At  present  we  may  assert  with  confidence  that 
the  procedure  to  secure  the  best  results  is  to  have  separate 
instruments  for  visual  and  for  photographic  work,  each  de- 
signed for  its  special  end.  There  is  no  doubt,  however,  that 
much  worth  having  may  be  gained  by  a  careful  study  as  to 
the  best  attainable  form  with  the  materials  to  be  used  —  a 
precaution  too  often  neglected. 


CHAPTER  VI 
THE  MICROSCOPE 

THE  history  of  the  microscope  is  far  less  easily  traced  than 
that  of  the  telescope.  This  is  chiefly  because  its  physical 
theory  is  so  much  more  complicated  than  that  of  the  latter 
instrument.  So  true  is  this  that  its  development  to  a  state  of 
almost  ultimate  perfection  preceded  all  adequate  theory  of  its 
action.  It  is  easy  to  point  out  certain  definite  discoveries  or 
inventions  in  the  course  of  development,  which  may  serve  for 
fixing  the  epochs  from  which  the  history  could  be  most  logi- 
cally traced ;  but  since  in  nearly  every  instance  these  discov- 
eries have  been  made  quite  empirically  and  utilized  is  trade 
secrets,  it  is  impossible  to  establish  their  date  accurately,  or 
even,  in  many  cases,  to  determine  to  whom  the  credit  of 
discovery  belongs.  We  shall  meet  with  the  names  of  a  few 
philosophers,  however,  who  have  done  something  toward  the 
improvement  of  this  important  instrument,  and  who  have 
given  the  results  of  their  labors  to  the  world  by  publication. 
Such  are  Lister,  Goring,  Amici,  Wenham,  and  finally  Abbe, 
who  has  not  only  brought  the  theory  of  the  microscope  to  a 
satisfactory  state  of  completion,  but  has  also,  in  conjunction 
with  the  manufacturers  Zeiss,  made  a  number  of  most  notable 
improvements  which  will  form  topics  for  consideration  some- 
what later.  Besides  these,  there  is  a  host  of  most  skilful 
artisans  who  for  half  a  century  have  carried  on  a  vigorous 
rivalry  in  the  attainment  of  increased  excellence.  Some  of 
these  are  notable  for  more  than  the  admirable  products  of 
their  skill  which  have  materially  advanced  our  knowledge 
of  nature,  for  to  them  are  due  for  the  most  part  those  inno- 


84  LIGHT 

vations  in  the  principles  of  construction  which  have  given  a 
new  impulse  to  the  art.  Such  are  Amici  in  Italy ;  Chevalier, 
Oberhaeuser,  and  Hartnack  in  France ;  Andrew  Ross  in  Eng- 
land ;  Spencer  and  Tolles  in  this  countiy. 

We  shall  give  here  a  brief  sketch  of  the  development  of 
the  microscope  in  regard  to  its  optical  improvement,  disre- 
garding the  hardly  less  interesting  history  of  its  mechanical 
construction  and  of  its  accessories. 

The  property  of  yielding  a  magnified  image  of  an  object, 
possessed  by  a  lens-formed  body,  was  without  doubt  known  to 
the  ancients.  At  least,  we  know  that  they  had  the  necessary 
skill  in  grinding  and  polishing  transparent  gems,  and  there 
remain  to  this  day  many  examples  of  antique  cameos  of  such 
exquisite  detail  in  finish  as  to  force  the  conclusion  that 
the  artist  must  have  been  aided  by  a  magnifying  glass. 
Nor,  considering  the  extreme  meagreness  of  references  to 
technical  methods  in  mechanical  arts  by  classical  writers, 
is  the  absence  of  all  contemporary  allusions  to  such  aids 
to  be  regarded  as  significant.1  But  however  true  the  conclu- 
sion may  be  in  regard  to  this  knowledge  of  the  ancients,  we 
do  know  that  not  until  after  the  middle  of  the  seventeenth 
century  did  the  microscope  add  greatly  to  our  knowledge  of 
nature.  It  is  highly  interesting  that  the  marvellous  scien- 
tific activity  of  this  century  opened  two  great  realms  for 
study  —  that  of  the  boundlessly  great  and  that  of  the  invis- 
ibly small.  Not  less  interesting  is  the  wide  difference  be- 
tween the  rates  at  which  exploration  in  these  new  fields  was 
pushed.  At  the  beginning  of  the  nineteenth  century  the 
truths  attainable  as  a  result  of  mere  increase  of  telescopic 
range  of  vision  had  been  almost  completely  gathered,  while 
those  made  accessible  to  us  by  the  microscope  were  only 
fairly  begun.  The  reason  for  this  disparity  in  progress  in 
the  two  fields  will  appear  implicitly  as  we  trace  the  history 
of  the  latter  instrument. 

1  Natural  lenses  formed  by  hardened  drops  of  gums  and  resins  must  have 
always  been  familiar  objects,  and  amber  was  a  substance  frequently  finished 
with  curved  surfaces  in  ancient  times.  It  seems  quite  incredible  that  the  magni- 
fying properties  of  such  bodies  could  have  been  unknown. 


THE  MICROSCOPE  85 

Almost  the  first  discoveries  of  importance  made  with  the 
microscope  were  published  in  a  famous  book  by  Robert 
Hooke,  in  1665,  entitled  "  Micrographia. "  The  book  is 
admirably  illustrated  by  numerous  engravings  on  copper  and 
bears  a  quaint  dedication  to  King  Charles  II.,  which  sug- 
gests many  reflections  regarding  the  changes  in  the  character 
of  scientific  publications  since  that  time.  The  dedication 
reads  as  follows :  — 

To  THE  KING 

SIR,  —  I  do  here  most  humbly  lay  this  small  Present  at  Your 
Majesties  Royal  feet.  And  though  it  comes  accompany 'd  with 
two  disadvantages,  the  meanness  of  the  Author,  and  of  the  Sub- 
ject; yet  in  both  I  am  incouraged  by  the  greatness  of  your 
Mercy  and  your  Knowledge.  By  the  one  I  am  taught,  that 
you  can  forgive  the  most  presumptuous  Offenders :  And  by  the 
other,  that  you  will  not  esteem  the  least  work  of  Nature,  or 
Art,  unworthy  your  Observation.  Amidst  the  many  felicities 
that  have  accompani'd  your  Majesties  happy  Eestauration  and 
Government,  it  is  none  of  the  least  considerable,  that  Philos- 
ophy and  Experimental  Learning  have  prospered  under  your 
Royal  Patronage.  And  as  the  calm  prosperity  of  your  Reign 
has  given  us  the  leisure  to  follow  these  Studies  of  quiet  and 
retirement,  so  it  is  just,  that  the  Fruits  of  them  should,  by  way 
of  acknowledgement,  be  returned  to  your  Majesty.  There  are, 
Sir,  several  other  of  your  Subjects,  of  your  Royal  Society,  now 
busie  about  Nobler  matters :  The  Improvement  of  Manufactures 
and  Agriculture,  the  Increase  of  Commerce,  the  Advantage  of 
Navigation :  In  all  which  they  are  assisted  by  your  Majesties 
Incouragement  and  Example.  Amidst  all  those  greater  Designs, 
I  here  presume  to  bring  in  that  which  is  more  proportionable  to 
the  smallness  of  my  Abilities,  and  to  offer  some  of  the  least  of 
all  visible  things,  to  that  Mighty  King,  that  has  establisht  an 
Empire  over  the  best  of  all  Invisible  things  of  this  World,  the 
Minds  of  Men. 

Your  Majesties  most  humble 
and  most  obedient 

Subject  and  Servant, 

ROBERT  HOOKE. 


86  LIGHT 

Hooke's  observations  were  made  chiefly  with  a  compound 
microscope  of  his  own  design  and  construction,  which  is  repre- 
sented in  the  accompanying  figure  copied  from  his  engrav- 
ing. The  lenses  consisted  of  a  small  one  at  the  lower  end  of 


FIGURE  21. 

the  tube,  which  served  to  form  a  magnified  image  of  an 
object  held  at  its  focus ;  an  eye  lens  near  the  top  of  the  tube 
with  which  to  view  the  image  formed  by  the  objective, 
and,  when  he  was  willing  to  sacrifice  something  of  the  dis- 
tinctness of  vision  for  an  increase  of  area  seen,  a  third  lens 
between  the  two  mentioned.  These  three  lenses  or  their 


THE  MICROSCOPE  87 

representatives  are  still  characteristic  of  the  compound  micro- 
scope. The  small  lens  which  Hooke  called  the  object-glass 
is  now  represented  by  a  system,  often  very  complex,  of  from 
two  to  ten  different  lenses,  acting,  however,  like  a  single  lens 
and  called  the  objective.  The  other  two  lenses  are  regarded 
as  a  combination,  styled  the  ocular.  Of  these  the  one  next 
the  eye  is  called  the  eye  lens,  and  the  other,  the  function 
of  which  is  to  increase  the  area  or  extent  of  field  viewed, 
is  called  the  field  lens.  In  modern  instruments,  however, 
nothing  is  to  be  gained  by  removing  the  field  lens. 

Hooke  distinctly  states  that  simple  microscopes  or  single 
lenses  of  high  power  yield  better  vision  than  his  compound 
instrument,  but  the  difficulty  of  securing  a  satisfactory  illumi- 
nation of  the  object  under  a  lens  of  very  short  focus  more  than 
counterbalanced  their  optical  superiority.  Hooke 's  method 
of  illumination  by  means  of  a  lamp  and  a  glass  globe  filled 
with  water,  which  acts  as  a;  so-called  condensing  lens,  is 
made  clear  by  the  figure,  and  it  is  perfectly  descriptive  of  the 
method  used  to  this  day  for  opaque  objects. 

Shortly  after  the  appearance  of  the  Micrographia,  Leeuwen- 
hoek,  a  Hollander,  commenced  the  publication  of  a  long  series 
of  microscopic  observations,  which  extended  from  1673  to 
1695.  In  scientific  importance,  though  not  in  popular  interest, 
these  papers  far  transcend  the  earlier  writings  of  Hooke, 
and  were  all  founded  upon  discoveries  made  with  simple 
lenses  of  his  own  grinding.  Of  these  Leeuwenhoek  left  sev- 
eral hundreds,  mostly  mounted  for  use  and  accompanied  by 
an  object  for  observation.  His  highest  powers  magnified 
about  270  times.  Twenty-six  of  these  finished  microscopes 
are  still  in  the  possession  of  the  Royal  Society  of  London,  to 
which  Leeuwenhoek  left  them  in  his  will.  A  figure  showing 
the  construction  of  these  is  given  on  the  following  page. 

At  this  point  the  improvement  of  the  microscope  as  an  opti- 
cal instrument  was  arrested  for  more  than  a  century.  In  out- 
ward form  it  underwent  continuous  modification,  so  that  in 
purely  mechanical  construction  the  compound  microscope  of 
the  beginning  of  the  nineteenth  century  is  not  greatly  unlike 


88 


LIGHT 


the  best  examples  of  the  modern  instrument;  but  leaving 
out  of  account  the  question  of  greater  convenience  in  use, 
its  absolute  power  as  an  instrument  of  research  was  less 
than  that  possessed  by  the  simple  microscopes  made  and  used 
by  Leeuwenhoek. 

We  are  thus  confronted  by  the  fact  that  while  the  ratio  of 
the  improvement  in  the  telescope  during  the  eighteenth  century 
—  taking  the  best  telescopes  of  Huyghens  and  Campani  as 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


FIGURE  22. 

the  measure  of  excellence  for  the  earlier  period  and  those 
of  Herschel  and  Schroeter  for  the  later  —  may  be  reckoned  at 
something  between  four  and  six,  and  the  ratio  since  1825  at 
one  and  a  half  or  two,  the  corresponding  numbers  for  the 
improvement  of  the  microscope  may  be  fairly  represented  by 
one  (equivalent  to  a  stationary  condition)  for  the  longer 
period,  and  as  four  for  the  interval  since  1825.  This  extraor- 
dinary inequality  in  development,  perhaps  unique  in  the  his- 
tory of  instrumental  aids  to  scientific  research,  offers  a  curious 


THE  MICROSCOPE  89 

subject  for  study.  Happily  the  elementary  theory  of  optical 
instruments  given  in  the  preceding  pages  is  quite  sufficient 
to  enable  us  to  follow  it  and  to  give  a  rational  explanation 
of  many  phenomena  not  generally  understood.  But  it  will 
carry  us  even  further  than  this ;  for  by  its  means  we  shall  find 
a  physical  limit  to  the  power  of  a  microscope  depending  upon 
the  very  nature  of  light  itself,  and  thus  shall  be  able  to 
attain  a  view  of  what  future  advances  in  the  art  may  be 
expected  to  bring. 

Consider  first  the  simple  lens  used  as  a  microscope,  and 
strive  to  find  the  ultimate  limit  of  its  power,  either  practical, 
such  as  may  be  determined  by  its  minuteness,  or  theoretical. 

The  action  of  a  magnifying  glass  is  explained  on  page  22. 
Its  office  is  to  form  a  virtual  image  of  an  object  very  near 
its  principal  focus  at  a  conveniently  greater  distance ;  or,  in 
other  words,  to  render  convex  wave-surfaces,  having  their 
centre  at  a  point  in  the  object,  flat  wave-surfaces  after  pass- 
ing the  lens.  We  shall  suppose  that  this  may  be  done  with- 
out error,  in  which  case  the  magnification  may  be  increased 
indefinitely  by  increasing  the  power  of  the  lens.  Any  in- 
crease of  power,  however,  demands  increased  curvature  of 
the  lens  surfaces  and  diminished  distance  between  the  lens 
and  the  object.  For  example,  a  lens  of  glass  of  spherical 
form,  which  to  the  unassisted  eye  would  make  an  object 
appear  one  hundred  times  larger  than  it  would  at  a  distance  of 
ten  inches,  would  require  the  object  to  be  within  one-thirtieth 
of  an  inch  of  the  surface  of  the  glass.  If  the  magnification 
were  ten  times  greater,  this  distance,  called  the  working  dis- 
tance of  the  microscope,  would  be  reduced  to  one  three-hun- 
dredth of  an  inch.  It  is  obvious  that  this  fact  would  put  a 
practical  limit  to  the  useful  power  on  account  of  the  diffi- 
culties of  illumination  and  adjustment.  In  special  cases  the 
working  distance  might  be  considerably  increased  by  using  a 
material  of  greater  refractive  power  than  glass,  such  as 
diamond,  sapphire,  or  garnet;  and  in  the  early  part  of  the 
century  many  experiments  were  made  with  these  substances, 
but  without  the  gain  hoped  for  by  their  advocates. 


90  LIGHT 

By  the  absolute  length  of  the  waves  of  light,  however,  there 
is  a  much  more  effective  limit  set  to  increase  of  magnification. 
For  it  is  self-evident  that  since  this  element  depends  upon 
increased  curvature  of  the  refracting  surfaces,  to  increase  the 
power  of  a  lens,  we  must  in  the  end  decrease  its  diameter; 
hence  the  diameter  of  the  wave-surfaces  after  passing  the  lens 
must  ultimately  be  less  with  greater  magnification.  Now  on 
page  38  it  was  shown  that  vision  through  a  hole  much  less 
than  one-sixteenth  of  an  inch  in  diameter  becomes  notably 
impaired,  because  such  dimensions  are  not  very  great  com- 
pared to  the  length  of  a  wave  of  light.  From  this  it  follows 
that  the  minute  details  of  an  image  could  not  be  fully  recog- 
nized if  the  diameter  of  the  lens  were  much  less  than  one- 
sixteenth  of  an  inch.  This  assertion  may  be  rendered 
convincing  if  we  consider  two  points  in  the  object  very  near 
each  other.  The  image  of  each  of  these  points  will  appear  as 
a  disk  of  a  determinate  size,  and  if  the  apparent  separation  of 
the  points  is  not  greater  than  the  diameter  of  these  disks  they 
cannot  be  seen  as  two  points,  but  as  one  only.  Increasing  the 
power  of  the  lens  does  not  help  in  the  least,  for,  although  it 
increases  the  apparent  separation  of  the  images,  it  at  the  same 
time,  on  account  of  necessarily  diminished  diameter,  increases 
in  exactly  the  same  ratio  the  diameter  of  the  disk  which 
represents  the  image  of  a  point  on  the  retina. 

The  experiments  just  referred  to  also  prove  that  when  the 
aperture  of  the  pupil  is  reduced  to  one-thirtieth  of  an  inch 
the  indistinctness  due  to  the  cause  under  discussion  becomes 
very  obvious ;  consequently  a  lens  smaller  than  this  in  diam- 
eter can  no  longer  add  to  the  power  of  vision,  since  each  point 
in  the  image  appears  as  a  disk  and  each  line  as  a  stripe,  of 
which  the  diameter  and  thickness  increase  directly  with  the 
magnification.  But  the  highest  power  which  a  lens  of  one- 
thirtieth  of  an  inch  in  diameter  can  have  is  600, 1  which  may 

1  The  proof  of  this  statement  is  given  in  At>pendix  A.  It  is  tacitly  assumed 
that  there  is  air  somewhere  between  the  object  and  lens ;  the  modification  neces- 
sary in  case  of  an  optically  denser  substance  surrounding  the  object  will  be  suffi- 
ciently clear  from  the  discussion  of  immersion  objectives. 


THE  MICROSCOPE  91 

therefore  be  stated  as  the  theoretical  limit  of  power  for  a 
simple  microscope.  This  conclusion  is  quite  independent  of 
the  material  of  which  the  lens  is  made.  In  every  case  the 
practical  limit  would  probably  be  found  considerably  below 
this ;  at  least  it  is  tolerably  certain  that  no  discoveries  have 
ever  been  made  with  simple  microscopes  magnifying  more 
than  250  to  300  times. 

In  the  case  of  a  simple  microscope  the  difficulties  arising 
from  too  close  an  approximation  of  the  eye  and  object  to  the 
lens  can  be  obviated  by  the  compound  microscope;  for  in 
this  case  the  eye  is  removed  by  a  little  more  than  the  length 
of  the  tube  from  the  object,  while  a  deficiency  of  power  in 
the  objective  can  be  compensated  by  increase  of  power  in  the 
ocular.  Thus  there  is  no  necessary  relation  either  between 
the  working  distance,  or  the  power  of  the  objective,  and  the 
total  magnification.  It  is  this  feature  which  led  Hooke  to 
prefer  the  compound  instrument  to  the  optically  superior 
single  magnifier.  However,  notwithstanding  the  ease  with 
which  the  magnifying  power  could  be  increased,  it  was  found 
that  nothing  was  gained  by  increasing  it  beyond  200  diameters. 
This  remained  true  until  later  than  1825.  The  cause  of  this 
failure  in  higher  powers  was  not  understood,  but  it  was 
attributed  to  defects  of  refraction  at  spherical  surfaces,  which 
we  know  under  the  names  of  spherical  and  chromatic  aberra- 
tions. Even  Dolland's  brilliant  discovery  of  a  method  of 
eliminating  the  latter  error  failed  to  bring  any  improve- 
ment in  the  microscope  for  more  than  seventy  years  after  its 
publication,, 

To  comprehend  the  cause  of  this  stationary  period  of  more 
than  a  century  in  the  development  of  the  compound  micro- 
scope, as  well  as  to  trace  intelligently  its  later  progress  to  a 
condition  approaching  perfection,  we  must  study  more  criti- 
cally than  appears  in  Chapter  II.  the  theory  of  the  instru- 
ment. To  do  this  we  shall  gain  much  by  considering"  the 
construction  from  a  somewhat  artificial  standpoint.  Instead 
of  regarding  the  objective  as  the  lens  of  a  camera,  as  in  that 
chapter,  and  the  ocular  as  a  magnifying  glass  by  means  of 


92  LIGHT 

which  the  image  formed  by  the  objective  is  observed,  we 
shall  regard  the  instrument  as  a  simple  microscope  placed 
just  in  front  of  a  small  telescope.  From  this  standpoint  the 
objective  forms  an  enlarged  image  of  the  object  at  its  prin- 
cipal focus,  which  is  at  a  great  distance  beyond  the  object, 
and  this  image  is  looked  at  through  the  telescope.  It  must 
not  be  concluded  that  this  artificial  method  of  consideration 
is  necessary  for  the  establishment  of  the  general  laws  which 
we  shall  derive,  but  it  is  convenient  because  the  physical 
theory  of  the  telescope  has  already  been  developed,  and  by 
this  means  we  shall  escape  the  necessity  of  establishing  any 
new  principles. 

The  magnifying  power  of  the  compound  microscope  thus 
regarded  becomes  equal  to  the  power  of  the  simple  micro- 
scope, which  constitutes  the  anterior  system  multiplied  by  the 
power  of  the  telescope.  Thus,  as  far  as  merely  geometrical 
considerations  are  involved,  it  is  a  matter  of  indifference 
whether  the  desired  magnification  is  secured  by  the  one  or 
the  other  of  the  two  parts.  Let  us  suppose  for  the  present 
that  the  portion  which  acts  as  a  simple  microscope  is  abso- 
lutely perfect,  leaving  until  later  the  consideration  of  a 
departure  from  this  assumption ;  then  the  simple  microscope 
forms  a  perfect  image  of  the  object  at  an  infinitely  great  dis- 
tance, the  apparent  size  of  which  is  lOp  times  the  apparent 
size  of  the  object  when  held  ten  inches  from  the  eye,  p 
being  its  power.  For  example,  if  the  power  of  the  lens  is 
1,  that  is,  if  it  has  a  focal  length  of  one  inch,  the  above 
value  is  10;  if  the  focal  length  is  one-half  an  inch,  giving 
a  power  of  2,  the  value  becomes  20,  and  so  on.  This  is 
a  merely  numerical  extension  of  what  has  been  stated  on 
page  21. 

In  the  chapter  concerning  the  telescope  it  was  shown  that 
the  highest  useful  power  is  about  thirty  times  the  number  of 
inches  in  the  diameter  of  the  objective.  In  the  case  under 
discussion  the  available  diameter  of  the  telescope  is  obvi- 
ously the  same  as  the  diameter  of  the  back  surface  of  the 
lens  or  lenses  acting  as  a  simple  microscope.  Let  this 


THE   MICROSCOPE  93 

diameter  be  represented  by  2FN,  F  being  the  focal  length 
of  the  simple  microscope  and  .ZVa  number  to  be  determined 
later;  then,  since  the  total  magnification  is  equal  to  that  of 
the  simple  microscope  multiplied  by  that  of  the  telescope,  the 
highest  useful  power  is  equal  to  lOp  x  30  x  2FN,  or  to 
6002V,  as  appears  from  the  fact  that  p  and  F  are  reciprocals  of 
each  other. 

The  largest  possible  value  for  N,  when  there  is  air  any- 
where between  the  object  and  the  front  of  the  microscope,  is 
1 : 1  consequently  in  the  class  of  instruments  under  discus- 
sion the  useful  magnification  is  limited  to  600  times.  The 
two  obvious  and  interesting  features  of  this  conclusion  are, 
first,  that  the  ultimate  useful  power  attainable  with  a  com- 
pound microscope  with  an  air  objective  is  the  same  as  that  of 
a  simple  microscope ;  second,  that  the  ultimate  useful  power 
of  a  compound  microscope  depends  upon  the  ratio  of  the 
effective  diameter  of  the  rear  surface  of  the  objective  to  its 
focal  length,  and  not  at  all  upon  the  power  of  the  objective 
nor  upon  the  length  of  the  instrument.  This  last  deduction 
recalls  the  somewhat  similar  rule  governing  the  ultimate 
useful  power  of  a  telescope. 

The  true  efficiency  of  a  perfect  microscope  is  thus  deter- 
minable  from  the  value  of  N  alone,  .which  should  therefore 
be  used  to  characterize  an  objective.  It  is  called  the  numer- 
ical aperture,  and  it  is  generally  represented  by  microscopists 
by  the  symbol  N.A.  Why  this  awkward  term  should  have 
been  chosen  by  Professor  Abbe,  to  whom  we  owe  its  intro- 
duction, will  appear  later. 

Before  proceeding  with  the  history  of  the  evolution  of  the 
modern  microscope,  we  may  perhaps  define  more  closely  the 
term  "highest  useful  power,"  particularly  as  it  leads  us 
directly  to  an  expression  for  the  maximum  defining  power  of 
a  given  type  of  microscopes.  It  will  be  observed  that  the 
number  600  in  the  foregoing  expression  for  power  depends 

1  The  proof  of  this  important  statement  is  given  in  Appendix  A;  it  consti- 
tutes the  foundation  of  the  physical  theory  of  the  microscope. 


94  LIGHT 

upon  our  arbitrary  assumption  of  a  limit  of  30  diameters  to  the 
inch  aperture  in  the  useful  power  of  a  telescope,  conse- 
quently, it  may  be  argued  that  the  limit  of  power  thus  deter- 
mined for  the  microscope  is  also  arbitrary.  This  is  true,  but 
not  important.  What  is  meant  is,  that  with  an  absolutely 
faultless  microscope  with  a  dry,  or  air,  objective,  all  the 
details  of  an  object  visible  with  any  power  whatever  could  be 
seen  with  a  magnification  of  600,  and  that  no  one  would  find 
any  advantage  in  employing  a  higher  power ;  most  observers, 
indeed,  would  prefer  400  or  500.  For  example,  let  us  sup- 
pose that  we  have  at  command  a  perfect  dry  objective  of  one- 
inch  focal  length  and  greatest  possible  value  of  JV,  then  the 
available  aperture  of  the  telescope,  which  combined  with 
this  objective  makes  a  compound  microscope,  is  two  inches. 
As  a  result  from  both  theory  and  practice,  however,  we  find 
that  the  closest  points  or  lines  which  can  be  seen  through  a 
telescope  have  an  angular  separation  of  4".56  divided  by  the 
number  of  inches  in  its  aperture  (see  page  72),  that  is,  for  this 
case  2". 28.  Now  an  arc  of  2".28  is  2.28/206265  of  the 
length  of  the  radius  with  which  it  is  described;  consequently 
this  fraction  represents  the  portion  of  an  inch  which  must 
separate  the  two  points  or  lines  that  can  be  just  seen  as  not 
single  by  means  of  an  objective  of  one-inch  focal  length.  The 
fraction  is  equal  to  1/90000  part  of  an  inch;  hence,  since  the 
separating  power  has  been  shown  to  depend  upon  the  value 
of  N  only,  we  may  make  the  general  deduction  that  the 
finest  structure  which  can  be  resolved  in  white  light  by 
means  of  an  air  objective  is  represented  by  90, 000  repetitions 
per  inch.  If  blue  light  is  used,  the  defining  power  increases 
in  the  inverse  ratio  of  the  length  of  the  light  waves.  For 
example,  if  light  of  the  same  wavelength  as  that  of  the  line 
F  in  the  solar  spectrum  is  employed  for  illumination,  the 
number  rises  to  100,000,  which  may  be  regarded  as  the  maxi- 
mum for  vision.  For  a  photographic  plate,  however,  the 
ultimate  defining  power  must  be  somewhat  increased  over 
this  limit,  possibly  twenty  per  cent. 

As  to  the  magnification  necessary  to  exhibit  this  fineness 


THE  MICROSCOPE  95 

of  structure,  we  have  the  following  considerations  to  guide 
us:  A  good  eye  will  just  recognize  a  system  of  lines  sepa- 
rated by  intervals  of  60  to  70  seconds  of  arc;  but  30  times 
2.28"  is  68.4";  hence  a  magnification  of  this  amount  in  the 
telescope  for  the  example  chosen,  corresponding  to  a  total 
power  of  300,  will  just  enable  a  keen  eye  to  see  the  struc- 
ture, while  twice  this  value  will  certainly  quite  reach  the 
limit  imposed  by  the  length  of  light  waves  even  with  inferior 
eyes.  This,  then,  is  the  meaning  of  the  ultimate  useful 
power  being  6QON. 

Until  the  importance  of  the  quantity  JVwas  recognized,  or 
at  least  suspected,  rapid  improvement  in  the  microscope  was 
impossible.  While  deficiency  in  power  was  attributed  to 
spherical  and  chromatic  aberrations  alone,  without  any  effort 
being  made  to  investigate  the  real  importance  of  these  defects 
in  absolute  measure,  nothing  was  to  be  expected  in  the  way 
of  a  rational  progress.  This  was  the  condition  of  the  affair 
throughout  the  eighteenth  century.  It  is  true  that  during 
that  time  a  method  of  eliminating  one  of  the  supposed  barriers 
to  progress  was  discovered,  namely,  the  principle  of  achro- 
matic compensation ;  and  this  was  early  applied  to  the  micro- 
scope, but  without  any  improvement  whatever. 

It  seems  to  have  been  an  accident  dependent  upon  the 
mechanical  difficulties  of  construction,  which  finally  let  in 
light  upon  the  hidden  principles  involved.  It  had  been 
noticed  as  a  practical  fact  that  it  was  by  no  means  a  matter 
of  indifference  whether  the  magnification  was  obtained  by 
a  powerful  objective  or  by  means  of  the  remainder  of  the 
optical  system,  but  that  powerful  objectives  always  yielded 
relatively  better  results.  When  achromatic  combinations 
were  tried,  however,  it  was  found  impossible  to  make  very 
powerful  lenses  on  account  of  their  minuteness,  the  power  of 
the  combination  being  only  the  difference  of  the  powers  of 
the  positive  crown  lens  and  the  negative  flint  lens  combined 
with  it.  But  in  1824  Selligue  had  the  happy  thought  of 
combining  a  number  of  binary  achromatic  lenses  so  as  to  act 
as  a  single  lens  of  increased  power  and  thus  retain  a  diameter 


96  LIGHT 

of  the  system  which  was  as  great  as  that  of  a  single  achro- 
matic. His  objective,  constructed  by  Vincent  and  Charles 
Chevalier,1  consisted  of  four  like  binary  lenses  which  could  be 
used  singly  or  combined  as  a  system  of  two  or  more.  The 
focal  length  of  each  was  about  one  and  a  half  inches.  The 
following  year  the  Chevaliers  improved  this  construction  by 
inverting  each  binary  lens  so  that  the  flat  sides  are  turned 
toward  the  object.  This  notable  improvement,  apparently  so 
insignificant,  was  followed  by  Professor  Amici  of  Modena,  in 
1827,  whose  success  is  emphasized  by  Lister.  Two  years 
later  Utschneider  and  Fraunhofer  were  supplying  objectives 
like  those  of  Amici,  except  that  the  powers  of  the  binary 
lenses  increased  from  the  rear. 

In  1830  a  very  remarkable  paper  by  Lister  appeared  in  the 
Transactions  of  the  Royal  Society,  in  which  he  showed  that 
the  relative  distances  of  the  binary  lenses  play  a  very  impor- 
tant role  in  the  function  of  the  objective.  Figure  23,  a,  shows 
the  form  of  his  objective,  which  is  still  universally  employed 
for  low  powers;  and  shortly  afterward,  by  an  extension  of 
the  principle,  the  optician  Tully  succeeded  in  producing  a 
triple  objective,  formed  of  three  binary  lenses  like  the  two 
figured,  which  possessed  a  value  of  N  equal  to  0.42.  Lister's 
paper  also  contained  the  description  of  an  apparatus,  the  first 
of  its  class,  for  determining  the  largest  angular  extent  of  a 
wave-surface  which  an  objective  can  transmit. 

This  paper  by  Lister  marks  an  epoch  in  the  history  of  the 
microscope.  Although  its  theoretical  importance  has  been 
greatly  over-rated  by  writers  on  the  microscope,  its  practical 
value  at  the  time  of  publication  hardly  admits  of  over-esti- 
mation. The  immediate  improvement  was  not  so  great  as 
that  made  in  Selligue's  objective,  since  objectives  with  a 
value  of  N  as  great  as  0.3  were  then  in  use  in  England; 
but  for  the  first  time  a  guiding  principle  was  laid  down  and 
a  method  given  for  determining  a  constant  that  bears  a 

1  Here,  as  in  all  following  diagrams  of  objectives,  the  object  is  supposed  to  be 
below  the  lower  line  of  the  figure.  The  shaded  areas  will  always  represent  sec- 
tions of  flint  lenses. 


THE  MICROSCOPE 


97 


simple  relation  to  the  true  measure  of  power  which  is  here 
indicated  by  N.  From  that  time  progress  toward  the  high 
degree  of  excellence  recently  attained  has  been  continuous. 
Dr.  Goring  had  already  introduced  the  practice  of  determin- 
ing the  efficiency  of  a  microscope  by  means  of  "test  objects," 
such  as  the  finely  lined  scales  of  various  insects  and,  with 


FIGURE  23. 

later  and  better  microscopes,  the  delicate  silicious  shells  of 
many  varieties  of  diatoms.  These  tests  and  Lister's  con- 
stant, which  soon  received  the  name  of  "angular  aperture,"1 

1  The  true  measure  of  efficiency,  which  has  been  indicated  by  N  in  the  preced- 
ing pages,  is  in  fact  the  sine  of  one-half  of  Lister's  constant.  It  was  doubtless  in 
order  to  distinguish  the  new  term,  and  at  the  same  time  to  suggest  its  relation 
with  the  old  one,  that  Professor  Abbe  employed  the  somewhat  infelicitous 
expression  "numerical  aperture." 

7 


98  LIGHT 

remain  to  this  day  the  ordinary  means  by  which  microscopists 
and  makers  characterize  their  objectives. 

Before  proceeding  with  this  sketch  of  the  history  of  the 
growth  of  the  microscope  it  is  well  to  note  the  work  done  in 
this  country  by  a  young  man,  Mr.  Edward  Thomas,  more 
particularly  since  it  has  been  quite  overlooked  by  writers  on 
the  history  of  this  instrument.  In  the  "American  Journal 
of  Science  "  for  1831  Mr.  Thomas  has  two  papers  describing, 
with  all  the  detail  necessary  to  guide  a  practical  optician  in 
their  construction,  two  objectives,  the  more  powerful  of 
which  is  represented  diagrammatically  in  Figure  23,  b.  This 
objective  had  a  numerical  aperture  of  0.6  to  0.7,  and  was 
doubtless  the  best  that  had  been  constructed  up  to  that  time 
—  a  supremacy  which  it  retained  for  a  period  of  ten  or  fifteen 
years.  Since  this  writer  seems  also  to  have  had  a  deeper 
insight  into  the  nature  of  the  difficulties  to  be  surmounted  in 
perfecting  the  microscope  than  his  contemporaries,  we  may 
fairly  conclude  that  his  death  in  this  same  year  was  a  real 
loss  to  science. 

Aside  from  this  forgotten  achievement  no  advance  in 
achromatic  objectives  is  recorded  between  1830  and  1837. 
At  least,  Goring  and  Pritchard  in  their  "  Micographia,"  pub- 
lished in  the  latter  year,  regarded  the  reflecting  microscopes 
constructed  on  Amici's  plan,  with  mirrors  made  by  Cuthbert, 
as  superior  to  any  achromatic  at  that  time  attainable.  Prob- 
ably no  man  then  living  was  a  better  critic  of  the  optical 
excellence  of  a  microscope  than  Goring,  and,  as  the  greatest 
value  for  N  with  his  instruments  was  0.46,  we  may  safely 
accept  that  number  as  a  measure  of  the  highest  efficiency 
then  reached.  It  is  worth  noting  that  only  at  this  epoch  did 
the  reflecting  microscope  ever  rival  the  refracting,  and  even 
then  Cuthbert  alone  was  able  to  make  the  essential  elliptical 
mirrors  of  such  great  aperture. 

In  1837  Andrew  Ross  introduced  the  practically  important 
improvement  of  so  mounting  the  lenses  that  the  distance 
between  the  front  and  middle  combinations  could  be  slightly 
changed  at  will,  thus  compensating  more  or  less  perfectly  the 


THE  MICROSCOPE  99 

altered  conditions  attending  different  thicknesses  of  cover- 
glasses.  This  adjustment  is  indispensable  in  dry  objectives 
of  large  aperture.  During  the  decade  following  this,  Ross 
and  Amici  were  leaders  in  improving  achromatic  objectives, 
although  at  the  end  of  this  period  Powell  and  Smith  in 
England,  Oberhaeuser  in  France,  and  Spencer  in  America 
were  almost,  if  not  quite,  on  an  equal  footing  with  their 
predecessors.  Oberhaeuser  and  his  successor  Hartnack  de- 
serve special  mention,  because  their  efforts  to  produce  efficient 
microscopes  at  a  relatively  small  cost  were  so  far  successful 
that  doubtless  more  serious  scientific  work  has  been  accom- 
plished with  instruments  from  their  factory  than  with  those 
from  any  other  source.  To  this  period  be- 
longs a  type  of  construction  for  objective- 
fronts,  introduced  probably  by  Ross,  which 

remained  for  some  time  the  prevailing  construc- 

.    &T^.  FIGURE  24. 

tion  for  high  powers.    It  is  shown  in  r  igure  24. 

Between  1850  and  1860  was  introduced  the  practice  of 
making  the  anterior  lens  of  high-power  objectives  of  a  single 
piece  of  crown  glass  nearly  hemispherical  in  shape.  Although 
the  value  of  this  innovation  stands  unapproached  in  the 
history  of  the  microscope,  excepting  possibly  those  of  Selligue 
and  of  Lister,  it  is  impossible  to  name  definitely  its  inventor. 
It  is  generally  attributed  to  Amici ;  but  Wenham,  whose  con- 
tributions to  practical  microscopy  have  been  important  in 
other  fields,  states  that  he  introduced  it  in  1850  and  exhib- 
ited it  in  successful  employment.  The  form  in  which  this 
improvement  was  rapidly  adopted  by  leading  makers  of  that 
period  is  shown  in  Figure  23,  c,  while  an  extension  in  the 
same  direction,  indispensable  for  the  highest  attainable 
powers,  and  probably  invented  by  the  skilful  American 
optician  Tolles,  is  exhibited  in  Figure  23,  d  ;  it  is  known  as 
the  double  front.  But  whosoever  the  discoverer  may  have 
been,  the  discovery  itself  was  not  only  a  necessary  step  in  the 
progress  toward  perfection,  but,  once  taken,  it  led  inevitably, 
by  a  rapid  approach,  to  modern  excellence.  • 

The  theory  of  the  hemispherical  front  is  not  difficult  to 


100  LIGHT 

comprehend.  The  office  of  every  lens  in  an  optical  instru- 
ment is  to  alter  the  curvature  of  the  wave-surfaces  trans- 
mitted by  it ;  if  the  transmitted  wave-surface  is  of  constant 
curvature,  that  is,  either  flat  or  strictly  spherical  in  form,  the 
lens  is  free  from  spherical  aberration ;  and  leaving  out  of  con- 
sideration the  chromatic  aberration  which  can  be  corrected 
by  other  means,  the  resulting  image  is  geometrically  perfect. 
Now  in  general  a  spherical  refracting  surface  does  not 
accomplish  this,  although,  with  the  present  mechanical  means 
at  command,  spherical  surfaces  alone  admit  of  precise  shap- 
ing. The  only  practical  method  of  attaining  the  required 
end  is  to  combine  convex  and  concave  surfaces  so  that  their 
errors,  opposite  in  kind,  shall,  after  the  method  of  Lister,  com- 
pensate each  other  as  far  as  possible.  The  resources  of  this 
method,  however,  are  exhausted  at  about  the  limit  of  N=  0.6, 
that  is,  further  progress  requires  either  a  new  principle  or  a 
method  of  making  accurate  lenses  of  other  than  spherical 
forms.  The  latter  means  appears  no  less  hopeless  now  than 
in  the  time  of  Descartes.  Fortunately  a  new  principle  was 
discovered  in  the  utilization  of  one  of  the  two  special  cases  in 
which  refraction  at  a  spherical  surface  is  geometrically 
perfect.  This  case  may  be  explained  by  reference  to  Figure 
25,  which  represents  a  piece  of  glass  bounded  at  the  left  by  a 
polished  spherical  surface  whose  centre  is  at  C.  A  line  Cp\ 
through  the  centre  is  chosen  as  the  axis  of  the  lens.  If  any  point 
of  this  line  more  remote  from  the  surface  than  0  is  a  source  of 
light  waves,  they  will  suffer  such  a  modification  in  passing 
from  the  glass  into  the  air  as  to  reduce  their  curvature, 
although  in  general  they  will  also  lose  their  spherical  form. 
It  is  obvious  that  if  the  source  corresponds  with  the  centre  (7, 
neither  the  curvature  nor  the  shape  will  be  changed ;  but  it  is 
also  true  that  there  is  another  point  p\,  from  which  spherical 
waves  will  be  refracted  without  loss  of  their  spherical  form, 
and  at  the  same  time  undergo  great  diminution  in  curvature. 
The  position  of  this  remarkable  point  is  defined  by  the  rela- 
tion Opi,  equal  to  the  radius  of  the  surface  divided  by  the 
index  of  refraction  of  the  glass.  A  very  simple  computation 


THE  MICROSCOPE  101 

will  show  that  the  image  of  this  point  pi  is  at  p%  so  placed 
that  Cpi  equals  the  radius  of  the  refracting  surface  multiplied 
by  the  refractive  index ;  consequently  the  ratio  of  Cp2  to 
Cp\t  which  is  also  that  of  the  magnification  of  the  image,  as 
is  obvious  from  the  figure,  is  equal  to  the  square  of  the  index 
of  refraction,  or  to  w2,  if  we  employ  the  ordinary  symbol  for 
that  constant. 


FIGURE  25. 

Here,  then,  is  a  means  by  which  an  object  inside  a  sphere 
of  glass  and  a  short  distance  from  the  centre  can  be  replaced 
by  a  virtual  image  w2  times  as  great  in  size,  and  absolutely 
without  faults  except  such  as  depend  upon  the  small  vari- 
ations of  n  for  different  wavelengths.  We  may  first  turn 
our  attention  to  the  limit  of  the  advantage  to  be  thus 
acquired,  and  afterward  consider  the  practical  methods  of 
securing  the  proper  physical  conditions. 

In  Figure  26  let  pz  represent  such  a  virtual  image  of  pu 
magnified  n2  times,  and  aob  an  indefinitely  thin  lens  system 
everywhere  equidistant  from  pz,  which  will  render  wave-sur- 


102  LIGHT 

faces  from  pz  exactly  flat.  It  is  evident  that  the  whole 
system  is  just  that  which,  combined  with  a  telescope,  will 
constitute  a  compound  miscroscope  in  the  sense  of  the  pre- 

vious analysis.  Moreover,  it 
must  be  admitted  that  this 
highly  artificial  construction 
is  at  least  as  efficient  as  any 
other,  because  by  assumption 
each  lens  performs  its  office 
perfectly.1  But  the  elements 
of  the  system  are  so  simple 
that  it  is  quite  easy  to  calcu- 
late the  ultimate  magnifying 
FIGURE  26  power.  Thus,  in  the  expression 

for  the  highest  useful  magni- 

fication on  page  93,  the  value  of  the  power  is  here  equal  to 
the  product  of  the  two  factors,  namely,  n2  due  to  the  first 
refracting  surface  and  ~L/op2  due  to  the  lens  system  ab  ; 
again,  the  aperture  of  the  telescope  is  equal  to  the  diameter 
of  ab,  or  to  2op2  sin  ap2o.  Substituting  these  values  in  the 
expression,  there  results  :  — 

600n  sin  ap2o. 

The  highest  possible  value  of  sin  ap2o,  as  appears  almost  di- 
rectly from  Figure  25,  is  \ln  ;  hence  the  value  of  the  ultimate 
useful  power  of  any  microscope  constructed  on  this  system  is 


N,  as  before,  being  a  number  never  greater  than  unity. 

From  this  analysis  it  appears  that  the  range  of  micro- 
scopic vision  is  only  limited  by  the  refractive  index  of  the 
material  of  which  the  front  lens  is  made.  Unfortunately 
very  high  refractive  power  seems  to  be  inseparable  from 
opacity,  and  we  know  nothing  now  which  can  approach 
diamond  as  a  possible  material,  though  it  would  obviously 
offer  great  mechanical  difficulties  in  use  :  its  index  of  refrac- 
tion is  about  2.5.  Many  other  substances  of  very  high 

1  See  Appendix  A  for  a  further  discussion  of  this  point,  and  also  of  the 
maximum  value  of  ab. 


THE  MICROSCOPE  103 

refractive  powers,  such  as  sulphur  and  phosphorus,  are 
rendered  impossible  in  use,  either  because  of  double  refract- 
ing property  or  of  their  mechanical  condition;  while  the 
densest  glasses,  with  indices  approaching  2,  have  such  large 
dispersion  that  their  employment  would  introduce  complica- 
tions most  difficult  and  perhaps  impossible  to  deal  with. 
Ordinary  crown  glass,  with  its  moderate  index  of  1.5,  appears, 
therefore,  to  be  the  best  substance  for  practical  use.  An 
objective  with  a  front  lens  of  such  a  material  and  the  object 
within  its  substance  should  enable  us  to  see  lines  as  separate 
objects  when  as  close  as  135,000  to  150,000  to  the  inch.  Since 
our  first  microscope-makers,  under  the  leadership  of  Pro- 
fessor Abbe  and  Dr.  Zeiss,  have  attained  within  five  or  ten 
per  cent  of  this  limit,  it  seems  not  extravagant  to  speak  of 
the  microscope  as  having  reached  practical  perfection. 

The  methods  by  which  in  practice  the  conditions  involved 
in  the  preceding  theory  are  secured  may  be  indicated  in  Fig- 
ure 27.  Here  the  hemispherical  fronts  of  three  objectives  are 


I 

FIGURE  27. 


represented  with  the  objects  to  be  viewed  below  them  under 
a  cover  of  thin  glass.  In  each  case  the  virtual  image  of  the 
object,  formed  by  all  the  successive  refractions  suffered  by 
the  wave-surfaces  up  to  that  by  the  spherical  surface  itself, 
must  be  very  near  the  point  corresponding  to  p^  in  Figure  25 ; 
but  the  continuity  of  the  glass  is  interrupted  by  a  thin  plate 
of  fluid  which  admits  of  the  necessary  adjustment  of  focus, 
the  fluid  being  air  in  a,  water  in  5,  and  in  c  a  liquid  as  simi- 
lar as  possible  in  its  optical  properties  to  those  of  crown 
glass.  The  introduction  of  these  layers,  save  in  the  last  case, 
causes  errors  in  the  refractions  which  become  greater  with 
greater  thickness  and  wider  difference  between  their  optical 
properties  and  those  of  glass.  Thus,  in  the  first  case,  where  air 


104  LIGHT 

is  the  dividing  medium,  the  layer  must  be  very  thin  if  a  large 
aperture  is  to  be  employed,  or  errors  will  be  introduced  which 
do  not  admit  of  correction  in  the  remainder  of  the  system ;  in 
short,  the  "working  distance"  of  an  objective  of  high  effi- 
ciency of  this  type  is  necessarily  very  small.  It  is  ordinarily 
called  a  dry  objective.  An  objective  constructed  with  a  front 
such  as  is  illustrated  by  5,  the  water  immersion  objective, 
possesses  much  greater  flexibility  in  respect  to  working  dis- 
tance, because  its  conditions  are  much  nearer  those  demanded 
by  theory.  Finally,  the  homogeneous  immersion  objective, 
often  called  the  oil  immersion  because  the  fluid  employed  is 
the  oil  of  red  cedar,  may  have  a  working  distance  up  to  nearly 
two-thirds  the  radius  of  the  hemispherical  front  and  still 
perform  satisfactorily. 

It  is  easy  to  show  that  the  expression  given  as  a  measure 
of  the  ultimate  power  of  the  homogeneous  immersion  objec- 
tive, namely,  BOOwiV,  would  apply  to  all  forms  if  n  repre- 
sents, not  the  refractive  index  of  the  substance  of  the  front 
lens,  but  that  of  the  least  refractive  medium  between  the 
object  and  the  spherical  surface  of  the  front  lens.  Thus,  for 
a  dry  objective  n  =  1,  for  a  water  immersion  n  =  1.33,  and 
for  a  homogeneous  immersion  n  =  1.5.  As  for  JV,  it  is  as 
easy  to  obtain  a  high  value  in  one  type  as  in  another,  and  the 
most  skilful  makers  succeed  in  constructing  excellent  objec- 
tives with  a  value  as  high  as  .93  to  .95.  From  these  data 
and  the  discussion  which  appears  on  page  93  we  may  conclude 
that  the  limit  of  resolution  of  dry  objectives  is  about  90,000 
lines  to  the  inch,  of  water  immersion  about  120,000,  and  of 
homogeneous  immersion  objectives  about  135,000.  It  is  an 
interesting  fact  that  in  order  to  secure  this  limit  in  the  type 
of  objectives  last  named,  it  is  necessary  to  employ  more  than 
a  full  hemisphere  in  the  front  lens,  as  is  evident  from  an  in- 
spection of  Figure  25.  This  obviously  heightens  enormously 
the  difficulty  of  shaping  and  mounting  such  lenses ;  hence  all 
objectives  of  the  greatest  possible  aperture  must  be  costly. 

No  review  of  the  microscope  at  this  period,  however  ele- 
mentary its  aims,  can  leave  unnoticed  Professor  Abbe's  very 


THE  MICROSCOPE 


105 


remarkable  invention  which  he  has  named  the  apochromatic 
objective.  To  give  a  rational  description  of  this  will  require, 
it  is  true,  a  step  further  in  the  theory  of  optical  instruments 
than  has  been  necessary  up  to  this  point,  although  it  is  a 
step  which  follows  very  naturally  what  precedes;  in  short, 
it  will  be  necessary  to  consider  some  of  the  chromatic  defects 
which  are  inseparable  from  the  old  construction.  For  this 
end,  suppose  the  objective,  which,  in  accordance  with  our 
previous  convention,  serves  only  to  make  a  perfect  virtual 
image  of  the  object  at  an  infinite  distance,  to  be  divided  into 

i 


FlGUR 

two  portions,  the  lower  of  which  (Z,  Figure  28,  A)  forms  a  mag- 
nified virtual  image  of  the  object  at  a  finite  distance,  and  the 
upper  part  it,  a  corrected  virtual  image  of  this  at  an  infinite 
distance.  But  since  all  transparent  substances  refract  short 
waves  more  than  long,  the  image  of  o  formed  by  I  will  be  a 
complex  of  colored  images  from  r  to  5,  the  blue  image  being 
the  most  remote.1  The  function  of  the  system  u  is  to  con- 

1  Of  course  the  system  I  is  undercorrected  for  color ;  if  not,  let  /  apply  to  the 
lowest  lens  nlone,  and  u  be  a  system  comprising  the  two  binary  lenses,  and  the 
reasoning  will  be  in  no  wise  changed. 


106  LIGHT 

vert  all  the  complex  system  of  wave-surfaces  having  their 
centres  distributed  from  r  to  b  into  plane  wave-surfaces ;  in 
other  words,  the  system  u  must  have  a  higher  power  for  red 
than  for  blue  waves,  and  at  the  same  time  be  sensibly  free 
from  spherical  aberration  for  all  colors.  There  is  no  diffi- 
culty in  meeting  these  conditions  for  wave -surf  aces  of  mod- 
erate angular  extent  by  a  combination  of  negative  lenses  of 
flint  glass  with  positive  lenses  of  crown  glass,  but  when  the 
wave-surfaces  are  large,  as  they  must  be  in  powerful  objec- 
tives, it  is  found  that  the  power  of  the  upper  system  is 
always  relatively  too  small  at  the  margin  for  the  short  waves 
of  light.  This  want  of  flexibility  in  the  means  of  correction 
arises  from  the  fact  that  in  all  known  glasses  a  great  increase 
in  dispersive  power  is  invariably  accompanied  by  increase  in 
refractive  power.  It  is  true  that  there  are  many  liquids  com- 
bining high  dispersive  power  with  relatively  small  refractive 
indices,  and  by  a  combination  of  such  a  liquid  with  crown 
glass  and  flint  glass  in  the  system  u,  Professor  Abbe  in  con- 
junction with  Dr.  Zeiss  succeeded  in  constructing  two  objec- 
tives of  unprecedented  excellence ;  but  such  fluid  lenses  are 
practically  quite  useless  on  account  of  the  mechanical  diffi- 
culties of  keeping  them  in  working  order. 

Fortunately  for  science  the  elimination  of  this  most  serious 
defect  in  high-power  objectives  did  not  depend  upon  the  dis- 
covery, not  very  promising  it  must  be  confessed,  of  a  variety 
of  glass  which  should  possess  the  optical  properties  of  some 
exceptional  fluids.  Professor  Abbe  devised  a  brilliant  expe- 
dient, which  seems  to  have  occurred  to  no  one  before  him, 
by  means  of  which  this  defect  is  eliminated  in  an  entirely 
different  manner.  Consider  the  changes  in  the  problem  if 
the  lens  system  u,  instead  of  standing  as  close  as  possible  to 
I,  is  removed  to  a  considerable  distance,  as  in  Figure  28,  B. 
Here  we  recognize  that  the  change  of  curvature  to  be  pro- 
duced by  u  is  lessened,  and  hence  that  the  power  of  the 
objective  as  a  whole  is  somewhat  decreased;  but  what  is 
vastly  more  significant  is  that  the  difference  of  curvature 
between  that  of  the  red  and  that  of  the  blue  is  decreased  in 


•    THE  MICROSCOPE  107 

a  much  greater  ratio.  Thus,  if  the  curvature  of  the  red  light 
waves  is  half  as  great  in  the  second  case,  this  difference  will 
be  only  one-fourth  as  great;  if  the  curvature  is  reduced  three 
times  by  making  the  distance  between  the  lens  systems  I  and 
u  twice  as  great  as  that  separating  I  and  r,  this  difference 
will  be  reduced  nine  times,  and  so  on.  The  modification 
consequently  admits  of  varying  the  relation  between  the  two 
distinct  functions  of  the  system  u  within  very  wide  limits, 
and  thus  yields  another  arbitrarily  variable  element  for  the 
attainment  of  a  closer  correction.  A  rational  use  of  this 
principle  with  a  skilful  choice  of  materials,  so  as  to  reduce 
to  a  minimum  the  far  less  serious  defect  of  what  is  called 
secondary  chromatic  aberration,  yields  the  highly  refined  apo- 
chromatic  objective. 

The  construction  of  the  apochromatic  entails  a  defect  of  a 
singular  character  in  the  complete  instrument,  that  is,  when 
used  as  a  part  of  an  ordinary  microscope,  which  is  of  inter- 
est on  account  of  its  general  nature.  In  Chapter  II.  it  has 
been  shown  that  the  magnification  of  a  virtual  image  depends 
upon  the  change  in  curvature  of  the  wave-surfaces  originat- 
ing in  the  object;  hence  the  lens  systems  I  of  Figure  28  magnify 
more  for  short  wavelengths  of  light  than  for  long,  and  conse- 
quently the  whole  objective  will  magnify  a  blue  object  more 
than  a  red  one  unless  the  systems  u  exactly  reverse  this 
relation,  that  is,  unless  they  change  the  curvature  of  the 
wave-surfaces  of  short  wavelength  less  than  those  of  longer 
waves,  and  less  by  just  the  proper  amount.  Now,  whatever 
may  be  the  case  with  the  system  u  in  Figure  28,  A,  it  is  clear 
that  the  corresponding  lens  system  in  Figure  28,  B,  cannot  cor- 
rect the  defect  in  question  in  the  anterior  portion,  because  the 
separation  has  been  made  for  the  express  purpose  of  reducing 
the  relative  difference  in  the  curvatures  of  the  wave-surfaces. 
Thus  it  appears  that  in  this  objective,  and  in  general  in  any 
lens  system  in  which  correction  for  color  is  obtained  by  lenses 
separated  by  a  considerable  distance  from  uncorrected  lenses, 
the  images  of  an  object,  although  all  in  the  same  plane,  and 
therefore  achromatic  in  the  ordinary  sense  of  the  word,  differ 


108  LIGHT 

materially  in  magnitude  with  differing  color.  This  defect 
produces  a  characteristic  imperfection  in  the  seeing,  with  all 
high-power  microscopes,  which  appears  as  a  form  of  color 
error,  increasing  very  rapidly  with  increasing  distance  from 
the  centre  of  the  field,  a  phenomenon  perfectly  familiar  to  all 
who  have  observed  critically  with  such  microscopes,  although 
it  is  not  generally  understood.  Professor  Abbe  corrects  this 
defect  by  adding  to  the  ocular  a  compound  lens  of  such 
construction  that  it  makes  its  power  greater  for  red  light 
than  for  blue  in  the  same  ratio  as  the  excess  of  magnification 
of  the  objective  lies  in  the  opposite  direction.  These  two  ele- 
ments combined,  namely,  the  apochromatic  objective  and  the 
compensating  ocular,  form  the  modern  perfected  microscope. 
In  the  theory  of  the  compound  microscope  there  is  still 
remaining  one  point  of  more  than  merely  technical  interest, 
and  that  is  the  question  of  distribution  of  power  between  the 
objective  and  the  ocular.  The  statement  has  been  previously 
made  that  the  experience  of  early  constructors  showed  that 
this  question,  though  not  suggested  by  theory,  was  a  most 
important  one.  They  found  that  in  general  much  better 
results  were  attained  by  the  employment  of  powerful  objec- 
tives and  weak  oculars.  The  reason  for  this  is  not  now  diffi- 
cult to  explain.  Ignoring  the  small  but  inevitable  errors  of 
construction,  we  have  just  learned  that  there  are  other  un- 
avoidable sources  of  imperfection  in  the  objective,  arising  in 
part  from  the  fact  that  errors  of  refraction  cannot  be  cor- 
rected where  they  occur,  but  only  in  more  or  less  remote 
portions  of  the  system,  which  must  bring  with  them  imper- 
fections in  the  images  to  be  enlarged  by  the  ocular.  How 
small  these  faults  can  be  made  is  a  question  of  experience 
rather  than  theory,  because  of  their  enormous  complexity  in 
kind  and  origin.  Such  experience  has  shown  that  with  the 
ordinary  type  of  powerful  objectives  the  images  are  so  far 
from  perfect  that  they  will  not  bear  a  magnification  of  more 
than  five  or  six  times  without  passing  the  limit  of  definition 
imposed  by  the  inherent  faults  of  construction ;  in  the  more 
perfect  apochromatic  objectives  this  magnification  may  be 


FIGURE  29. 


110  LIGHT 

raised  as  high  as  from  twelve  to  fifteen.  As  we  have  found 
that  the  highest  useful  power  of  a  dry  objective  is  about  600, 
we  conclude  that  a  good  objective  of  the  old  type  of  a  power 
of  100,  that  is,  of  Vio  inch  focal  length,  or  of  the  apochromatic 
construction  of  Vs  or  %  of  an  inch,  will  accomplish  all  that 
any  other,  however  powerful,  can.  In  a  similar  way  we  are 
led  to  conclude  that  the  older  type  of  homogeneous  immer- 
sion objective  of  Vis  inch  focal  length,  or  an  apochromatic 
of  Vio  inch,  are  quite  sufficient  to  exhaust  the  range  of 
microscopic  vision,  at  least  as  far  as  our  present  knowledge 
of  the  nature  of  light  and  of  vision  enables  us  to  declare. 

On  page  86  appears  a  figure  of  the  earliest  compound  micro- 
scope of  which  we  have  much  knowledge.  The  accom- 
panying picture  (Figure  29)  of  an  instrument  which  may  be 
regarded  as  one  of  the  best  existing  models  serves  as  an 
interesting  contrast.  It  is  provided  with  three  objectives  of 
different  powers,  any  one  of  which  may  be  brought  into  use 
at  once  by  a  rotation  of  their  common  carrier.  The  appa- 
ratus below  the  stage  is  for  the  purpose  of  securing  at  will 
illumination  from  any  desired  direction  and  angular  extent. 
It  is  known  as  Abbe's  illuminator. 


CHAPTER  VII 
OPTICAL  PHENOMENA  OF  THE   ATMOSPHERE 

THERE  are  many  phenomena,  some  beautiful,  some  merely 
curious,  which  depend  upon  the  modifications  that  light 
undergoes  in  its  passage  through  the  atmosphere.  In  cer- 
tain cases  these  phenomena  are  necessary  consequences  of  the 
optical  properties  of  the  air  alone,  or,  at  least,  dependent 
upon  invariable  constituents  of  the  air,  as,  for  example,  the 
blue  color  of  the  sky,  mirages  and  looming,  and  scintillation 
of  the  stars;  in  other  cases  we  find  the  causes  in  bodies 
temporarily  suspended  in  the  atmosphere,  as  in  the  rainbow, 
corona,  and  halo. 

If  the  atmosphere  were  absolutely  transparent,  that  is,  if 
light  waves  could  be  regularly  transmitted  through  it  with- 
out loss,  we  should  find  the  sky  quite  black  except  where 
a  bright  spot  marked  the  presence  of  a  star  or  planet.  In 
short,  the  sky  of  day  would  differ  in  appearance  from  that  of 
night  only  by  the  presence  of  the  sun.  On  the  other  hand, 
if  the  air  should  transmit  but  a  portion  of  the  wave-energy, 
converting  the  remainder  into  heat  or  some  other  form  of 
energy  inappreciable  by  the  eye,  we  should  have  no  closer  a 
resemblance  to  the  sky  of  experience ;  it  would  still  remain 
black  except  at  points  in  the  direction  of  stars  a  part  of 
whose  light  would  reach  the  eye.  As  a  matter  of  fact, 
Professor  Langley  has  demonstrated  that  what  we  call  a 
clear  atmosphere  is  only  slightly  opaque  to  light  waves.  The 
atmosphere  is  then  neither  a  perfectly  transparent  body  nor 
an  imperfectly  transparent  one.  What  it  is  we  may  possibly 
picture  to  ourselves  most  quickly  and  accurately  for  our 
purpose  in  the  following  way:  Imagine  a  perfectly  trans- 


112  LIGHT 

parent  atmosphere  —  therefore  a  black  sky  —  with  a  high 
sun;  now  imagine  distributed  throughout  this  atmosphere 
small  drops  of  water,  say  one-thousandth  of  an  inch  in 
diameter.  If  these  are  infrequent,  perhaps  one  to  each  thou- 
sand cubic  feet  of  space,  the  sky  would  send  light  to  the  eye 
from  all  directions,  since  each  drop  would  scatter  the  light 
which  fell  upon  it;  notwithstanding  this  action,  the  bright- 
ness of  the  direct  sunlight  might  not  be  notably  diminished. 
Such  a  sky  woul'd  be  called  a  hazy  sky.  If  the  number  of 
drops  should  be  continuously  increased,  the  density  of  the 
haze  would  grow  while  the  brightness  of  the  direct  sunlight 
would  progressively  diminish  until  the  sky  became  entirely 
overcast  and  the  sun  invisible.  With  the  exception  of  the 
initial  black  sky,  just  such  a  change  has  been  repeatedly 
observed  by  every  one.  Now  reverse  the  process,  that  is, 
suppose  a  portion  of  the  drops  to  be  removed,  leaving  the 
distribution  uniform,  until  the  sky  is  no  brighter  than  a 
clear  sky  as  we  ordinarily  observe  it;  then  suppose  the  drops 
to  be  reduced  in  size  but  at  the  same  time  increased  in  num- 
ber at  such  a  rate  that  the  total  quantity  of  light  from  the 
sky  remains  unchanged.  This  reduction  in  size  has  intro- 
duced an  entirely  new  element  into  the  consideration,  for 
when  this  is  carried  so  far  that  the  particles  of  water  have  a 
diameter  which  is  small  compared  to  a  wavelength  of  light, 
they  are  no  longer  capable  of  reflecting  these  waves,  precisely 
as  a  floating  body  on  the  ocean  would  be  incapable  of  reflect- 
ing waves  whose  lengths  are  great  compared  to  its  own 
dimensions,  although  it  would  be  a  perfect  barrier  to  the 
passage  of  short  waves.  Before  reaching  this  condition  of 
extreme  tenuity,  however,  we  should  have  passed  through 
a  range  of  dimensions  which  might  be  called  small  when 
measured  by  the  lengths  of  red  light  waves,  but  not  small 
when  measured  by  lengths  of  blue  or  violet  waves.  Such 
particles  would  reflect  violet  light  ancl  blue  light  more  copi- 
ously than  orange  and  red  lights.  This  is  the  explanation  of 
the  blue  color  of  the  sky. 

It  is  immaterial  what  the  particles  are  made  of,  provided 


COLOR   OF  THE  SKY  113 

that  they  are  sufficiently  small  and  not  coagulated;  hence, 
when  we  receive  light  diffused  from  a  cloud  of  smoke  which 
is  not  too  dense,  especially  if  the  background  is  black  so  that 
no  other  than  this  diffused  light  reaches  the  eye,  we  see  a 
similar  pale  blue.  The  blue  of  the  opal  and  of  opalescent 
bodies  has  its  origin  from  a  similar  cause,  as  has  also  the 
color  of  blue  eyes  and  of  the  deep  sea.  It  is  well  known  to 
painters  that  generally  a  mixture  of  a  black  with  a  white 
paint  gives  a  strongly  bluish  gray,  although  there  may  be  no 
suggestion  of  this  color  in  either  of  the  components.  This, 
too,  is  explained  by  this  species  of  selective  reflection  depend- 
ing upon  minuteness  of  reflecting  particles. 

From  the  preceding  considerations  it  follows  that  sunlight 
which  has  come  to  us  through  the  atmosphere  has  lost  more 
in  short  than  in  long  waves;  consequently  the  hue  of  such 
light  is  somewhat  yellow.  If  the  light  has  passed  a  very 
long  distance  through  the  air,  as  when  the  sun  is  near  the 
horizon,  we  may  have,  together  with  a  very  great  diminution 
in  the  strength  of  all  waves,  a  practically  complete  stoppage 
of  the  short  waves.  This  would  leave  yellow,  orange,  or 
red,  depending  on  the  completeness  of  the  action,  and,  also 
(as  we  shall  see  when  we  come  to  study  the  phenomena  of 
color  sensations),  to  a  considerable  extent  upon  the  absolute 
intensity  of  the  light.  The  colors  are  strongest  after  the 
sun  is  below  the  horizon  and  sends  light  to  us  only  by  the 
medium  of  reflecting  clouds,  for  then  the  path  of  the  light 
through  the  air  is  much  longer  than  when  the  source  is 
above.  The  bluish  greens  and  blue-greens  which  are  not 
infrequently  seen  in  a  sunset  sky  are  often  a  physiological 
effect  of  contrast. 

Quite  an  analogous  phenomenon  to  the  last  is  presented  by 
a  light  smoke  which  appears  blue  against  a  dark  background, 
but  yellow  when  the  background  is  such  as  to  send  much 
more  light  to  the  eye  than  the  smoke  itself. 

One  of  the  most  noteworthy  effects  of  this  peculiar  opacity 
or  opalescence  of  the  atmosphere  is  the  change  it  produces 
in  the  aspect  of  distant  objects.  Thus  two  surfaces,  the  one 

8 


114  LIGHT 

light  and  the  other  dark,  lose  something  of  their  contrast  as 
they  recede  from  the  eye,  the  first  becoming  darker  by  the 
absorption  of  its  light  by  the  intervening  air  and  the  second 
brighter  from  the  superadded  light  diffused  by  the  air.  If 
the  air  is  free  from  relatively  large  particles  of  water  and 
from  coarse  dust  particles,  this  added  light  is  blue,  and  it 
often  requires  only  a  moderate  distance  to  give  a  strongly 
blue  hue  to  shadows  on  a  sunlit  rock,  while  distant  hills  are 
always  of  a  strong  violet  or  blue  tint  in  a  clear  day.  This 
effect  goes  under  the  general  term  of  aerial  perspective,  and 
it  affords  the  readiest  means  of  estimating  the  distances  of 
remote  objects  on  land.  In  an  exceptionally  clear  and  dry 
atmosphere  the  effect  is  greatly  diminished,  whereas  a  moist 
climate  enhances  the  impressiveness  of  mountain  scenery. 
A  light  fog  or  haze  often  lends  a  charm  to  landscapes  which 
otherwise  would  be  quite  uninteresting. 

There  is  another  property  of  the  atmosphere,  which,  though 
sufficiently  obvious  to  the  astronomer,  is  ordinarily  over- 
looked, at  least  in  its  common  manifestations,  by  one  whose 
attention  has  not  been  especially  directed  to  it.  The  veloc- 
ity of  light  waves  in  air  of  the  prevailing  density  at  the 
surface  of  the  earth  is  about  three  parts  in  ten  thousand  less 
than  in  a  vacuum.  From  this  and  the  decreasing  density  at 
higher  altitudes  it  follows  that  when  light  enters  obliquely 
into  the  atmosphere  its  course  is  changed  by  refraction  to  an 
amount,  when  the  obliquity  is  greatest,  nearly  equal  to  the 
diameter  of  the  sun.  The  sun,  therefore,  appears  to  be  just 
above  the  horizon  when,  were  there  no  atmosphere,  it  would 
appear  to  be  just  below  it.  The  effect  is  to  lengthen  the 
day  at  the  equator  by  about  four  minutes,  though  at  higher 
latitudes  this  lengthening  would  be  greater,  even  rising  to 
many  hours  in  extreme  latitudes.  This  atmospheric  refrac- 
tion is  attended  by  the  secondary  phenomenon  of  dispersion, 
as  are  other  cases  of  refraction,  but  it  requires  a  powerful 
telescope  to  detect  this  fact,  and  it  is  consequently  of  minor 
interest. 

Far  more  striking  than  the  regular  atmospheric  refractions 


WOLLASTON'S  EXPERIMENT  115 

are  the  curious  phenomena  known  as  looming  and  mirage, 
which  find  their  cause  in  temporary  inequalities  of  atmos- 
pheric density.  The  complexity  of  these  phenomena  is  enor- 
mous, nor,  except  in  most  general  terms,  have  they  been 
adequately  explained ;  but  after  establishing  an  optical  princi- 
ple of  great  importance,  we  shall  find  it  quite  easy  to  deduce 
a  consistent  explanation  from  an  ingenious  experiment  to 
illustrate  their  origin,  invented  by  Wollaston. 

The  principle  named  may  be  stated  as  follows :  At  any 
point  in  a  wave-surface  which  has  its  origin  in  a  distant  body, 
imagine  a  small  portion  cut  out,  say  in  the  form  of  a  circle ; 
then,  if  the  light  is  modified  in  any  way  so  that  this  circle  is 
changed  to  any  other  shape  or  size,  the  object  seen  from  the 
point  named  will  appear  changed  in  a  manner  exactly  recip- 


FlGURE   30. 

rocal  to  this.1  For  example,  if  the  circle  a,  in  Figure  30, 
represents  the  portion  cut  out  of  the  unmodified  wave- 
surface,  in  which  certain  points  are  numbered  for  conven- 
ience in  tracing  the  changes,  and  the  forms  which  follow  rep- 
resent respective  modifications  produced  by  various  optical 
means,  then  the  first  one  (5)  shows  that  the  object  is  changed 
from  right  to  left,  that  is,  that  it  is  perverted  as  it  would  be  by 
reflection  from  a  vertical  mirror ;  the  next  (V)  implies  a  per- 
version in  a  vertical  plane ;  d,  in  turn,  shows  that  the  object 
appears  inverted  and  magnified  in  the  ratio  of  the  diameter 
of  the  large  circle  to  the  small  one ;  in  short,  it  is  this  principle 
which  was  used  on  page  70  for  determining  the  magnifying 
power  of  a  telescope  ;  finally,  e  and  /  indicate,  respectively,  a 

1  The  proof  of  this  important  principle  is  given  in  Appendix  A,  section  vi. 


116  LIGHT 

magnification  in  the  vertical  plane,  and  an  equal  diminution 
without  change  in  the  horizontal  direction. 

We  now  turn  to  Wollaston's  experiment.  This  consists  of 
a  glass  tank  with  parallel  sides  (he  used  a  square  bottle, 
although  it  is  difficult  to  find  one  sufficiently  regular  to  show 
all  the  phenomena  to  be  described),  into  which  he  poured  first 
a  layer  of  transparent  syrup  ;  upon  this  a  layer  of  water  was 
placed  so  as  not  to  disturb  the  syrup ;  finally,  a  layer  of  alco- 
hol was  placed  with  like  precaution  upon  the  water.  These 
three  liquids  are  perfectly  miscible,  but  are  placed  in  order  of 
their  density,  so  that  they  mix  only  by  the  slow  process  of 
diffusion.  Their  optical  densities,  however,  follow  a  quite 
different  order,  the  middle  layer  being  less  refractive,  that  is, 
producing  less  retardation  of  the  wave-surfaces,  than  either 
the  syrup  or  the  alcohol.  If  now  a  distant  object  in  the 
same  horizontal  plane  as  the  tank  is  looked  at  through  the 
liquid  from  a  point  quite  close  to  the  tank,  the  following 
phenomena  may  be  noted :  Starting  from  a  level  where  the 
syrup  may  be  regarded  as  unmixed  with  water,  one  sees,  first, 
an  erect  image  of  the  object  in  its  true  position  and  size ;  as 
the  eye  is  raised  the  object  is  lifted  above  its  true  position 
and  increased  in  vertical  dimension  ;  from  a  still  higher  point 
the  object  is  lifted  still  more,  but  recovers  its  original  size ; 
beyond  this  point  follows  a  less  elevation  with  a  diminution 
in  vertical  dimension,  and,  when  the  eye  is  at  a  level  where 
only  pure  water  is  in  the  line  of  vision,  the  object  is  again 
seen  in  its  true  position  and  magnitude.  As  the  eye  is 
gradually  raised  to  the  level  of  the  pure  alcohol,  the  whole 
series  of  phenomena  is  reversed,  not  only  as  regards  position, 
but  also  in  respect  to  elongation  or  compression  in  the  verti- 
cal direction.  If  the  series  of  observations  is  repeated,  with 
the  eye  remote  from  the  tank,  perhaps  eight  or  ten  feet,  much 
more  striking  effects  will  be  noted.  In  the  first  place  the 
change  in  level  of  the  eye  necessary  to  bring  all  the  phases 
into  view  will  be  found  to  be  much  larger.  Secondly,  there 
suddenly  appears,  at  a  considerable  height  above  the  unmodi- 
fied image  of  the  object,  a  second  image,  which  quickly  betrays 


WOLLASTOWS  EXPERIMENT  117 

itself  as  a  double  image,  the  lower  of  the  two  being  inverted. 
As  the  eye  rises,  the  under  one  of  these  two  images  leaves 
the  upper  and  approaches  the  lowest  of  the  three,  with  which 
it  finally  unites  and  ultimately  vanishes.  For  some  distance 
above  this  point,  depending  upon  the  thickness  of  the 
stratum  of  water  which  may  be  regarded  as  unmixed  with 
either  syrup  or  alcohol,  only  one  image  will  be  seen  which 
is  unchanged  in  size  only  when  undeviated;  in  every  case 
diminution  in  the  vertical  plane  accompanies  deviation.  A 
further  raising  of  the  eye  carries  one  through  a  reversed  repeti- 
tion of  the  phenomena.  The  ratio  of  vertical  height  of  the 
inverted  images  to  that  of  the  erect  ones  varies  with  the 
distance  of  the  eye  from  the  tank,  and  is  not  especially  sig- 
nificant for  our  purposes  ;  but  when  this  distance  is  the  least 
that  admits  of  a  good  inverted  image,  it  will  be  found  to  be 
magnified  in  the  vertical  direction.  What  is  of  more  interest 
to  us  in  its  bearing  upon  the  explanation  of  mirage  and  other 
extraordinary  refractions  is  the  fact  that  only  from  below  the 
lower  transition  stratum  and  from  above  the  upper  transition 
stratum  can  multiple  images  be  seen ;  in  other  words,  recalling 
the  optical  character  of  the  fluids  used,  only  from  that  side 
of  the  transition  strata  upon  which  the  more  refrangible  fluid 
is  found.1  The  explanation  of  the  first  series  of  phenom- 
ena will  appear  very  simple  from  a  consideration  of  Figure 

1  The  experiment  herewith  described  is  as  simple  to  carry  out  as  it  is  interest- 
ing. A  tank  having  a  distance  of  four  inches  between  its  plate-glass  ends  will  be 
found  of  a  convenient  size.  Into  this  may  be  poured  a  dilute  solution  of  sugar  ; 
upon  this  float  a  thin  piece  of  wood  or  cork,  and  pour  slowly  a  proper  depth  of 
water  upon  the  float.  This  procedure  will  secure  a  separation  of  the  two  fluids. 
If  desired,  the  alcohol  may  be  introduced  afterward  in  a  similar  manner.  In 
order  to  attain  the  most  satisfactory  results  the  vessel  should  remain  undisturbed 
for  a  number  of  hours,  to  secure  a  sufficiently  gradual  change  in  density  in  pass- 
ing from  one  region  to  another.  In  addition  we  may  note  that  too  great  a  differ- 
ence of  refractive  power  is  likely  to  be  found  if  the  syrup  is  not  rather  dilute,  but 
if  two  solutions  only  are  used,  this  is  very  readily  adjusted  by  thoroughly  mixing 
and  replacing  a  part  of  the  mixture  by  pure  water.  Also  a  gain  in  distinctness 
will  be  found  by  observing  through  a  horizontal  slit  of,  say,  a  fiftieth  of  an  inch 
in  width,  as  this  reduces  the  effect  of  astigmatism  due  to  the  cylindrical  form  of 
the  wave-surfaces. 


118 


LIGHT 


31.  Here  the  line  AB  represents  the  position  of  the  tran- 
sition layer  which  is  defined  as  the  region  where  the  rate  of 
change  of  optical  density  is  a  maximum,  and  the  lines  CD 
and  C'D'  the  position  of  layers  below  and  above  which, 
respectively,  the  density  may  be  regarded  as  constant.  A  flat 
wave-surface  entering  from  the  right,  since  it  travels  faster  in 
the  region  above  the  line  AB,  and  since  there  is  no  discon- 


ul 
\ 

SN 

i 

1 

i 

1 

A 

T  :'\  " 

FIGURE  31. 

tinuity  in  the  fluid,  will  take  successively  the  forms  repre- 
sented by  ws,  wfsf,  wnslf.  When  it  emerges  from  the  tank,  it 
will  be,  in  general,  slightly  modified  both  in  direction  and  in 
curvature,  but  as  this  change  will  always  be  very  small  and 
in  no  case  will  lose  its  line  of  inflection,  which  is  repre- 
sented by  the  straight  part  of  w>'V,  we  may  ignore  this  modi- 
fication in  our  explanation.  If  the  eye  is  placed  at  a  point 
near  Z>,  where  the  wave-surface  is  flat  and  perpendicular,  the 
point-source  of  the  wave,  and  therefore  the  surface  of  the 
object  in  its  immediate  neighborhood,  will  appear  in  its  true 
place,  unchanged  in  magnitude,  just  as  if  seen  through 
any  plane  and  parallel  plate.  But  as  the  eye  is  raised  toward 
B,  the  object  will  seem  to  rise  above  its  true  position,  since 
the  source  of  a  wave-surface  will  always  appear  in  a  direction 
at  right  angles  to  this  surface  and  at  the  same  time 
stretched  out  in  the  vertical  direction,  as  is  manifest  from 
the  optical  principle  given  above.  A  little  higher  the  vertical 
magnification  will  become  less,  since  the  curvature  of  the 
wave-surface  grows  less,  but  the  elevation  will  increase  until 


WOLLASTON'S  EXPERIMENT 


119 


the  eye  reaches  the  level  B,  where  the  elevation  is  a  maxi- 
mum, yet  the  dimensions  are  unchanged  because  the  wave- 
surface  is  there  again  flat.  As  the  eye  moves  from  this  point 
upward,  the  elevation  of  the  object  seems  to  decrease,  but  it 
is  subject  to  a  deformation  of  a  contrary  character,  that  is,  it 
will  appear  to  be  reduced  in  its  vertical  dimensions,  this 
reduction  attaining  a  maximum  where  the  wave-surface  has 
its  greatest  curvature,  and  finally  vanishing  with  the  elevation 
at  the  point  D'.  If  higher  up  there  were  another  transition 
layer  of  an  opposite  kind,  such  as  would  obtain  if  a  stratum 
of  alcohol  were  poured  upon  the  water,  all  the  phenomena 
described  would  ensue  in  a  reverse  order.  We  need  not, 
however,  stop  longer  over  this  particular  series  of  observa- 
tions, since  the  very  few  atmospheric  phenomena  here  imitated 
can  be  better  discussed  when  we  turn  to  that  series  presented 
to  the  eye  remote  from  the  tank. 

We  shall  find  it  advantageous  for  our  purposes  to  consider 
separately  the  two  cases  of  the  transition  layer  lying  above 
the  less  dense  medium  (alcohol-water)  and  above  the  denser 
medium  (water-syrup).  Taking  these  in  the  order  named, 
we  shall  meet  first  with  those  phenomena  of  extraordinary 
atmospheric  refraction  which  are  most  familiar. 


FIGURE  32. 


In  Figure  32,  let   abed  represent  the   wave-surface  after 
modification  by  a  refractive  medium  which  possesses  a  sufn- 


120  LIGHT 

ciently  rapid  increase  of  optical  density;  this  wave-surface 
will  move  on  parallel  to  itself  until  it  may  be  represented  by 
the  line  a'b'c'd',  in  which  the  straight  portion  ab  remains 
unchanged,  but  the  concave  part  be  is  greatly  contracted, 
while  the  remaining  convex  portion  is  considerably  elongated. 
Still  later  the  surface  assumes  the  form  indicated  by  the  line 
allb",  or  the  straight  portion,  still  unmodified ;  bnclf,  formerly 
the  concave  portion,  now  rendered  convex  and  inverted,  and 
c"d",  which  remains  convex  as  before. 

This  diagram  is  all  that  is  necessary  to  explain  these  phe- 
nomena of  vision,  although  it  is  well  to  recognize  that,  in 
order  to  represent  quantitatively  the  alcohol-water  experi- 
ment, the  horizontal  scale  of  our  figure  should  be  increased 
a  hundred-fold  or  more ;  and  to  extend  it  to  the  cases  pre- 
sented us  by  occasional  atmospheric  conditions  a  thousand- 
fold might  be  none  too  little.  To  an  eye  in  the  region  1 
of  the  diagram,  only  a  single  image  of  the  distant  source  can 
be  seen,  namely,  that  clue  to  the  flat  portion  of  the  wave,  and 
consequently  unaltered  both  in  direction  and  magnitude. 
At  0,  however,  a  sudden  change  takes  place,  for  here  three 
portions  of  the  original  wave-surface  enter  the  eye,  the  straight 
portion  giving  an  undeviated  and  undistorted  image ;  a  de- 
pressed erect  image  belonging  to  the  convex  portion  of  the 
wave,  also  undistorted,  since  the  wave-surface  is  here  prac- 
tically flat ;  and,  finally,  an  inverted  image  superimposed 
upon  this,  likewise  undistorted.  As  the  eye  sinks  into  the 
region  3,  the  second  and  third  of  the  images  described  rise 
toward  the  first,  but  this  displacement  is  more  rapid  in  the 
case  of  the  inverted  image,  which  at  the  point  indicated  by 
3  is  midway  between  the  others.  It  is  evident  from  the  fig- 
ure that  the  erect,  depressed  image  is  always  smaller  in  the 
vertical  direction  than  the  undeviated  image,  but  that  the 
distortion  of  the  inverted  image  depends  upon  the  distance 
of  the  eye  from  the  region  marked  o,  so  that  if  this  distance 
is  too  small  the  vertical  magnification  will  become  indefinitely 
large  and  the  corresponding  image  will  not  be  seen.  This  is 
the  reason  why  in  the  experiment  it  is  necessary  to  remove  the 


WOLLASTON'S  EXPERIMENT  121 

eye  a  considerable  distance  from  the  tank,  in  order  to  see  an 
inverted  image.  Finally,  when  the  eye  is  carried  into  the 
region  marked  5,  only  a  single  image  will  be  seen  slightly 
depressed  and  shortened  vertically.  It  is  important  to  ob- 
serve that  the  multiple  images,  whether  double  or  triple,  are 
seen  only  when  the  eye  is  between  the  points  marked  2  and 
4,  both  of  which  are  above  the  level  of  the  transition  stratum  ; 
in  short,  we  may  state  the  rule,  which  will  prove  useful  later, 
that  only  when  the  eye  is  on  that  side  of  the  transition  stratum 
upon  which  the  more  refrangible  medium  lies  is  it  possible  to 
see  multiple  images.  There  remains  but  a  single  remark  to 
be  added  before  we  are  in  the  position  to  explain  the  most 
familiar  of  all  cases  of  extraordinary  atmospheric  refractions, 
namely,  that  if  the  eye  remains  fixed  in  position  and  the  dis- 
tant object  is  moved  downward,  it  will  undergo  successively 
all  the  changes  here  described.  It  is  easy  to  see  that  this  fol- 
lows, from  the  fact  that  such  a  change  simply  changes  the 
inclination  of  the  wave-surfaces  by  a  small  amount  (recall  in 
this  connection  that  the  horizontal  scale  of  the  figure  should 
be  increased  hundreds  of  times),  which  would  thus  leave  the 
form  of  the  surfaces  practically  unchanged.  Therefore,  if 
one  observes  a  surface  indefinitely  removed  from  the  tank,  all 
the  foregoing  phenomena  may  be  seen  on  this  surface  simul- 
taneously; and,  since  all  variation  of  the  refractive  action  is 
confined  to  the  vertical  direction,  the  resulting  modifications 
in  the  appearance  of  the  object  will  be  arranged  in  horizontal 
bands.  For  this  reason  a  coarsely  printed  card  at  a  consider- 
able distance  is  well  adapted  for  the  experiments  with  the 
tank. 

Although  atmospheric  air  never  has  mixed  with  it  any 
substance  which,  like  the  sugar  in  the  tank  experiment, 
modifies  its  refractive  power  materially,  still,  since  its  refrac- 
tive power  decreases  with  its  temperature,  it  is  quite  conceiv- 
able that  in  some  conditions  the  temperature  might  be  so 
distributed  that  a  similar  layer  arrangement  of  refractive 
power  might  ensue  so  as  to  yield  many  of  the  results  dis- 
cussed in  the  pages  immediately  preceding ;  and  indeed  this 


122  LIGHT 

is  the  fact.  For  example,  when  cold  air  lies  above  the 
relatively  warm  water  of  a  lake  or  sea  (a  state  arising 
from  the  appearance  of  a  cold  wind,  or,  more  frequently, 
from  nocturnal  cooling  of  the  air  by  radiation),  the  con- 
dition in  question  is  very  frequently  found.  Indeed,  in 
late  summer  or  early  autumn,  after  the  waters  of  our  lakes 
and  bays  have  been  accumulating  heat  for  a  long  time,  it 
is  a  rare  exception  when  such  phenomena  cannot  be  seen  in 
greater  or  less  perfection  during  the  cooler  portions  of  the 
day.  Under  such  circumstances  the  air  in  the  immediate 
neighborhood  of  the  water  is  warmer,  less  dense,  and  less 
refractive  than  the  air  at  higher  levels;  consequently  an 
object  at  a  distance  greater  than  our  sensible  horizon  and 
sufficiently  near  the  horizon  will  send  light  to  us  which  on 
a  part  of  its  path  has  been  subject  to  the  optical  modification 
illustrated  in  Figure  32.  It  is  true  that  this  light  does  not 
enter  the  region  where  it  suffers  this  extraordinary  modifica- 
tion, through  a  vertical  plate ;  but  this  fact,  although  it  com- 
plicates the  problem,  does  not  alter  its  essential  nature.  We 
shall  have,  therefore,  a  more  or  less  complete  representation 
of  the  phenomena  discussed  in  connection  with  that  figure. 
There  is,  however,  one  difference  of  material  moment  in 
the  atmospheric  analogue,  which  must  be  here  noted.  The 
warmer  air  below  is  not  only  less  refractive,  but  it  is  also 
specifically  lighter;  hence  it  is  not  in  equilibrium,  and  con- 
stancy of  the  optical  images  can  be  hardly  expected.  On 
the  contrary,  the  intense  unsteadiness  of  such  images,  which 
often  results  in  most  grotesque  changes  in  form,  is  one  of 
their  striking  peculiarities.  It  also  follows  from  this  insta- 
bility of  equilibrium  that  not  only  must  there  be  a  constant 
supply  of  heat  from  below  to  sustain  it,  but  a  true  transition 
stratum  is  never  present,  since  the  temperature  must  be 
highest  just  at  the  boundary  surface.  From  the  second  con- 
sequence it  obviously  follows  that  the  convex  portion  of  the 
wave-surface  is  wholly  missing,  and  hence  the  erect  image 
below  is  never  seen.  By  reference  to  Figure  32  it  is  now  easy 
to  describe  the  phenomena  which  would  follow  in  the  appear- 


MIRAGE  123 

ance  of  a  very  distant  point  when  the  eye  is  continuously 
lowered.  At  1  a  single  erect  image  would  be  seen,  but 
when  the  eye  falls  to  #,  a  second  image  will  suddenly  appear 
far  below  it ;  further  progress  will  be  attended  with  the  rise 
of  the  lower  image,  which  of  course  is  an  inverted  one,  until 
it  unites  with  the  undeviated  one  and  both  vanish  simulta- 
neously. It  is  not  difficult  to  extend  the  explanation  to  the 
case  where  the  eye  is  at  rest  and  the  distant  object  has  appre- 
ciable dimensions.  Suppose,  for  example,  that  the  distant 
object  is  a  ship  and  that  the  eye  is  in  the  region  1\  then  the 
upper  parts  of  the  masts  and  sails  would,  if  the  ship  is  tall 
enough,  be  seen  single ;  but  as  the  observer  directs  attention 
to  the  lower  portions  he  would  notice  that  at  a  certain  level 
an  inverted  repetition  appears  some  distance  below,  and  that 
the  intervening  space  is  occupied  with  the  erect  image  above 
and  the  inverted  below  which  comes  up  to  meet  it  much  as 
it  would  appear  to  do  were  the  ship  resting  upon  a  horizon- 
tal mirror,  except  that  there  would  be  a  region  of  indistinct 
vision  between  the  two.1 

From  what  is  said  in  the  last  paragraph  about  the  physi- 
cal condition  which  attends  the  production  of  such  a  layer 
arrangement  of  the  air,  it  is  quite  evident  that  these  par- 
ticular phenomena  can  be  seen  only  over  extensive  plane 
surfaces  which  are  kept  continuously  at  a  temperature  much 
higher  than  the  adjacent  air.  This,  however,  can  be  also 
brought  about  when  such  a  flat  surface  is  continuously  heated 
by  the  sun,  and  nothing  is  more  common  to  those  who  look 
for  them  than  such  effects  over  smooth  pavements  or  along 
sun-heated  walls.  But  there  is  another  instance  which  has 
been  known  for  an  indefinite  time  and  often  described  —  that 
of  the  desert  mirage.  Over  extensive  plains  thus  heated  by 

i  This  last  statement  will  become  evident  to  the  mathematician  who  reflects 
on  the  character  of  the  locus  of  the  centres  of  the  curve  be;  it  is  a  curve  having 
its  vertex  near  o,  but  running  out  asymptotically  to  both  of  the  right  lines  or" 
and  oh" '.  Moreover,  it  is  easy  to  see  that  this  image  in  the  neighborhood  of  the 
VI)  will  be  strongly  astigmatic,  whence  comes  a  very  obvious  vertical  streakiness 
here.  Often  this  is  the  first  appearance  of  the  approach  of  conditions  favorable 
for  observing  the  more  complex  phenomena. 


124  LIGHT 

the  sun  the  rarefied  air  adjacent  to  the  ground  gives  an  in- 
verted image  of  the  sky  near  the  horizon,  and,  if  there 
happens  to  be  an  object  beyond  the  sensible  horizon  which 
raises  itself  through  this  lowest  stratum,  it,  also,  will  appear 
to  be  reflected,  and  thus  in  a  surprising  degree  of  likeness 
imitate  the  effect  of  a  distant  and  quiescent  sheet  of  water. 
A  moderate  wind  enhances  the  effect  in  general,  because  it 
serves  to  keep  the  upper  air  at  a  uniform  temperature,  while 
it  will  be  so  checked  close  to  the  earth  by  friction  that  it 
does  little  toward  cooling  the  air  in  contact  with  the  ground; 
of  course  a  high  wind  is  unfavorable. 

Turning  now  to  the  case  where  the  lower  strata  of  air  are 
denser  than  those  aloft,  we  are  brought  to  the  consideration 
of  more  complicated,  more  interesting,  but,  at  least  in  our 
latitudes,  much  rarer  phenomena.  Here,  of  course,  there  is 
no  mechanical  unstability,  since  the  denser  layers  are  every- 
where below  the  less  dense ;  thus  we  shall  not  be  surprised  to 
learn  that  these  appearances  are  far  steadier,  and  afford  much 
better  optical  images.  It  is  well  to  note  that  the  preceding 
phenomena  as  well  as  those  which  follow  can  be  best,  and 
sometimes  only,  seen  by  aid  of  a  telescope  of  moderate 
power.  Even  an  opera  glass  is  an  efficient  aid. 

Figure  33  represents  this  case  diagrammatically.  Here 
an  eye  in  the  region  1  will  see  a  single  undeviated  image  of 
the  distant  object,  but  at  2  the  beginning  of  a  confused  image 
in  the  elevated  position  indicated  by  the  direction  of  the 
line  c"c.  This  latter  will  quickly  resolve  itself  into  an  erect 
image  above  and  an  inverted  image  below,  which  at  3  will 
be  about  midway  between  the  two  upright  images.  As  the 
eye  rises,  the  inverted  image  will  approach  the  lowest  one 
and  join  it  at  <£.  Above  this  position  there  will  be  but  a 
single  image  slightly  flattened  in  a  vertical  plane;  finally, 
above  6  all  evidence  of  extraordinary  refraction  will  disap- 
pear. The  condition  of  affairs  between  4  and  6  is  worthy  of 
a  moment's  attention.  Since  the  path  of  this  portion  of  the 
waves  is  concave  toward  the  earth,  sometimes  strongly  so, 
an  object  below  the  true  horizon  may  be  seen,  as  is  generally 


MIRAGE 


125 


true  of  a  rising  sun  or  moon,  and  many  mariners  report  hav- 
ing seen  and  recognized  ships  under  such  circumstances. 
The  extraordinary  flattening  of  the  rising  full  moon,  espe- 
cially in  hot  autumn  evenings,  is  often  noticed.  It  is  also 
the  explanation  of  the  famous  Fata  Morgana. 


FIGURE  33. 


The  favorable  meteorological  conditions  for  the  production 
of  this  class  of  phenomena  consist  in  the  existence  of  air  at 
a  high  temperature  over  a  cold  sea;  they  are  therefore  not 
often  seen  in  our  southern  waters,  but  over  the  colder  waters 
of  northern  bays  and  those  surrounding  England  they  are  less 
infrequent.  It  is  also  evident  that  any  circulation  of  the  air 
attending  a  wind  would  be  fatal  to  an  extensive  development 
of  the  appearances;  hence  one  can  hope  to  see  them  only 
during  calm  weather.  Thus  it  is  that  the  most  surprising 
accounts  of  like  phenomena  come  to  us  from  arctic  explorers 
who  have  had  opportunities  of  observing  over  a  partly  frozen 
sea  in  calm  and  warm  days  of  spring  or  summer.  Among  such 
observers,  Scoresby  was  the  first  who  published  a  carefully 
written  account  of  his  observations,  which  have  since  served 
as  the  basis  of  theoretical  discussions  by  numerous  writers. 
In  view  of  what  has  been  previously  said  concerning  the  scale 


126  LIGHT 

of  the  figures,  it  is  of  interest  to  note  that  this  observer 
never  found  well-developed  cases  when  the  object  was  less 
than  ten  to  fifteen  miles  distant,  which  makes  the  distance  of 
the  region  of  the  air  that  acts  like  our  tank  about  half  as 
much.  Almost  all  the  cases  which  he  pictures  are  in  per- 
fect agreement  with  the  conclusions  deduced  from  Figure  33, 
but  there  two  cases  of  apparent  disagreement  which  are 
worthy  of  further  consideration.  The  first  is  represented  by 
a  number  of  figures  in  which  the  erect  image  appears  below, 
with  an  inverted  image  above,  the  second  erect  image  which 
should  be  seen  surmounting  all  being  absent.  This,  how- 
ever, is  readily  explained  if  we  imagine  the  line  ddff,  which 
marks  the  lower  limit  of  the  region  of  uniform  density,  to 
be  lowered,  or,  what  amounts  to  the  same  thing,  the  part 
cd  of  the  wave-surface  to  be  much  shortened;  in  this  case 
the  image  in  question  would  be  much  diminished  in  height 
and  brightness,  and  might  well  escape  detection  if  the 
observer  were  not  guided  by  theoretical  knowledge.  The 
other  case  is  that  of  four  images  in  a  vertical  line.  This  may 
be  explained  by  the  assumption  of  the  existence  of  another 
transition  stratum  above,  an  assumption  which  does  not 
appear  at  all  improbable,  in  view  of  the  fact  that  such  an 
arrangement  would  be  in  stable  equilibrium.  Indeed,  it 
requires  considerable  care  to  avoid  the  production  of  just 
such  extra-transition  strata  in  the  syrup-water  tank;  while 
the  presence  of  two  enables  the  experimenter  to  copy  the 
case  of  four  images  very  satisfactorily. 

The  explanation  of  the  singular  phenomenon  known  as 
scintillation,  or  twinkling  of  the  stars,  has  given  a  vast  deal 
of  trouble  to  philosophers  of  both  ancient  and  modern 
times.  In  its  most  ordinary  manifestations  it  appears  as  a 
continuous  but  extremely  irregular  variation  in  the  apparent 
brightness  of  the  stars  when  not  too  near  the  zenith.  This 
appearance  is  generally  absent  in  the  case  of  the  planets,  even 
when  strongly  marked  in  the  fixed  stars,  but  it  may  be  seen 
sometimes  in  terrestrial  sources  of  light,  provided  that  they 
are  of  very  minute  angular  dimensions  and  sufficiently  re- 


SCINTILLATION  127 

mote.  When  the  phenomenon  is  quite  regular,  a  bright  star 
appears  to  undergo  continuous  change  of  color  with  only 
moderate  changes  of  brightness.  If  an  observer  presses  one 
eye  slightly  with  the  finger  so  as  to  see  two  images  of  a 
twinkling  star,  or  attains  the  same  result  by  placing  a  thin 
prism  before  one  of  the  eyes,  it  will  be  seen  that  the  changes 
in  one  of  the  images  are  quite  independent  of  those  in  the 
other.  Finally,  in  telescopic  images  of  scintillating  stars, 
at  least  in  telescopes  of  moderate  aperture,  quite  similar 
phenomena  are  visible  even  in  faint  stars.  These  may  be 
varied  in  an  instructive  way  by  shaking  the  telescope  slightly 
so  that  the  image  of  the  star  is  stretched  out  into  a  ribbon  of 
light,  in  which  case  the  curve  described  by  the  image  appears 
to  be  made  up  of  most  vividly  colored  elements.  For  a  long 
time  it  was  supposed  that  Arago  had  given  a  thoroughly 
satisfactory  explanation  of  the  whole  series  of  phenomena  by 
attributing  them  to  interference  of  light  waves  which  enter 
different  portions  of  the  pupil  or  objective,  and  which  must 
have  experienced  different  retardations  in  their  passage 
through  the  atmosphere ;  indeed,  for  more  than  two  genera- 
tions this  explanation  was  regarded  as  one  of  the  triumphs  of 
the  undulatory  theory  of  light.  Respighi  first  demonstrated 
the  inadequacy  of  Arago's  theory,  by  a  spectroscopic  exami- 
nation of  scintillating  stars,  the  results  of  which  he  embodied 
in  a  long  series  of  conclusions  of  which  the  most  important 
are  the  following :  — 

I.  In  spectra  of  stars  near  the  horizon  we  may  observe 
dark  or  bright  bands,  transverse   or  perpendicular  to   the 
length  of   the  spectrum,  which  travel  more  or  less  quickly 
from  the  red  to  the  violet  or  from  the  violet  to  the  red,  or 
oscillate  from  one  to  the  other  color;  and  this,  however  the 
spectrum,  may  be  directed  from  the  horizontal  to  the  vertical. 

II.  In  normal  atmospheric  conditions  the  motion  of  the 
bands  proceeds  regularly  from  red  to  violet  for  stars  in  the 
west,  and  from  violet  to  red  for  stars  in  the  east ;  while  in 
the  neighborhood  of  the  meridian  the  movement  is  usually 
oscillatory  or  even  limited  to  one  part  of  the  spectrum. 


128  LIGHT 

From  these  statements  it  is  at  once  evident  that  the  rota- 
tion of  the  earth  is  concerned  in  the  phenomenon,  and, 
although  the  previously  accepted  explanation  does  not  neces- 
sarily exclude  this  as  a  part  of  the  cause,  the  fact  that  it  did 
not  suggest  it  must  be  taken  as  indicating  a  lack  of  com- 
pleteness if  nothing  more.  But  the  interesting  and  unex- 
pected character  of  these  observations  led  Lord  Rayleigh  to 
study  critically  the  physics  of  the  problem,  and  to  recognize 
that  Arago's  premises  are  quite  untenable,  and  his  theory 
therefore  wholly  fallacious.  The  true  explanation  is  to  be 
looked  for  in  the  optical  irregularities  of  the  atmosphere, 
combined  with  the  obvious  fact  that  the  paths  of  the  short 
wavelengths  which  reach  the  eye  are,  on  account  of  the  not 
inconsiderable  dispersive1  power  of  air,  rather  widely  sepa- 
rated from  those  followed  by  the  longer  wavelengths.  In 
every  case  the  path  of  the  more  refrangible  light  lies  above 
that  of  the  less  refrangible.  If,  therefore,  there  is  a  limited 
volume  of  air  which  alters  the  form  of  a  wave-front  pass- 
ing through  it  so  that  less  of  that  particular  light  enters 
the  eye  at  a  stated  moment,  a  very  short  time  later  the  rota- 
tion of  the  earth  will  have  brought  this  air  to  a  higher  point 
if  it  lies  toward  the  west  and  to  a  lower  if  toward  the  east. 
Thus,  in  accordance  with  Respighi's  observations,  the  spec- 
trum of  a  star  in  the  east  would  show  progressive  changes 
from  violet  to  red,  while  in  an  opposite  quarter  of  the 
heavens  the  effect  would  be  reversed.  The  whole  series  of 
changes  would  occupy  but  a  short  time,  in  general  less  than 
a  second.  The  orderly  change  in  the  spectrum  of  a  star  on 
the  meridian  would  not  follow,  since  in  this  case  the  motion 
of  the  earth  does  not  carry  the  disturbing  body  of  air  from 
the  lower  level  of  the  paths  of  the  longer  wavelengths  to  a 
higher  one,  where  it  would  modify  the  more  refrangible  light. 
Since  a  very  small  angular  displacement  of  a  star  would 
entirely  alter  the  position  of  the  light-paths  with  respect  to 
a  small  body  of  air,  it  is  easy  to  see  why  different  points  of 
the  disk  of  a  planet  may  scintillate  in  so  perfectly  unre- 


CORONAS  129 

lated  a  way  that  the  light  from  the  planet  as  a  whole  does 
not  change  in  intensity.1 

Of  bodies  suspended  in  the  atmosphere  which  may  be  the 
source  of  important  optical  phenomena,  drops  of  water  and 
crystals  of  ice  only  are  of  sufficiently  frequent  occurrence  in 
our  climate  to  be  of  especial  interest.  To  the  first  we  owe 
coronas  and  rainbows ;  to  the  second,  halos  and  their  accom- 
panying appendages.  We  may  direct  our  attention  to  them 
in  the  order  here  named. 

Coronas  consist  of  one  or  more  colored  circles  concentri- 
cally surrounding  the  sun  or  moon  when  these  are  covered 
by  light  and  transparent  clouds.  They  are  readily  distin- 
guished from  the  larger  circles  which  are  called  halos,  not 
alone  on  account  of  their  smaller  and  variable  sizes,  but  also 
by  the  fact  that  the  inner  edge  is  blue  and  the  outer  red,  an 
order  which  is  reversed  in  halos.  Fraunhofer  showed  that 
these  coronas  may  be  exactly  imitated  by  scattering  very 
small,  circular,  opaque  bodies,  such  as  lycopodium  powder, 
in  a  perfectly  irregular  manner  over  the  surface  of  glass 
and  looking  through  such  a  plate  at  the  sun  or  moon.  By 
employing  a  telescope  so  as  to  magnify  the  effect,  and  by 
choosing  a  much  smaller  source  of  light  —  a  star  or  planet, 
for  example  —  he  was  able  to  secure  the  same  result  when  the 
powder  was  replaced  by  a  large  number  of  equal  disks  of 
tin  foil  scattered  in  a  perfectly  irregular  manner  in  front  of 
the  objective.  These  experiments  demonstrate  at  once  that 
we  have  to  do  with  an  effect  of  diffraction,  and  we  must 
therefore  look  to  the  laws  which  govern  this  class  of  phenom- 
ena for  a  rational  explanation.  It  is  necessary  to  add  little 
to  what  appears  in  Chapter  III.  to  make  the  matter  clear. 

In  that  chapter  it  was  shown  that  a  bright  point  seen 
through  a  very  small  round  hole  would  appear  as  a  disk  sur- 
rounded by  a  series  of  rings,  blue  on  the  inner  edges  and  red 
on  the  outer.  If  another  hole  of  the  same  size  were  perfo- 
rated in  the  screen,  we  found  that  the  disk  and  rings  remain, 

1  Further  considerations  with  reference  to  the  phenomenon  of  scintillation  may 
be  found  in  Appendix  B. 

9 


130  LIGHT 

but  are  doubled  in  brightness  and  crossed  by  a  series  of  dark 
lines.  If  the  number  of  holes  is  increased,  taking  care  to 
keep  them  of  the  same  size,  the  disk  with  its  concentric  rings 
remains,  but  with  greatly  increased  brightness ;  while  the  sys- 
tem of  dark  lines  has  gradually  become  so  complicated  and 
condensed  that  it  ultimately  becomes  indistinguishable.  Thus 
the  final  effect  would  be  the  same  as  that  of  a  single  hole  of 
the  size  chosen,  but  with  the  brightness  multiplied  by  the 
number  of  apertures.  This  conclusion,  the  validity  of  which 
has  been  mathematically  established  by  Verdet,  may  be 
readily  tested  by  pricking  a  large  number  of  holes  in  a  bit  of 
tin  foil  with  the  point  of  a  fine  sewing-needle,  taking  care 
to  secure  equality  of  size  by  thrusting  the  needle  through 
the  same  distance  each  time,  and  then  looking  through  this 
screen  at  an  artificial  star.  A  similar  effect  may  be  produced 
by  looking  at  a  bright  star  with  a  telescope  through  a  screen 
of  paper  perforated  with  irregularly  distributed  holes  even  as 
large  as  a  tenth  of  an  inch  in  diameter;  if  the  planet  Jupiter 
is  selected  as  the  object,  the  resulting  image  is  an  almost  per- 
fect imitation  of  a  well-developed  lunar  corona. 

To  make  use  of  the  facts  just  established  in  the  explana- 
tion of  coronas,  it  is  necessary  to  state  and  explain  a  fertile 
method  in  this  department  of  optics,  known  as  the  principle 
of  Babinet.  This  principle  may  be  thus  given:  If,  on  ac- 
count of  the  presence  of  an  opaque  screen,  however  compli- 
cated, we  have  light  at  any  point  between  the  source  and  that 
point,  then,  if  all  the  transparent  portions  of  the  screen  are 
made  opaque  and  the  opaque  portions  transparent,  the  quan- 
tity of  light  at  the  point  will  remain  unchanged.  The  proof 
of  this  fact  becomes  obvious  if  we  consider  Figure  14,  page  32. 
The  reason  why  light  is  found  at  pi  is  only  because  the 
wave-surface  is  limited;  consequently  the  opaque  portion 
of  the  screen  cuts  off  just  what  would,  if  added  to  the  dis- 
turbances which  reach  this  point,  exactly  destroy  them ;  in 
other  words,  the  waves  suppressed  by  the  screen  are  of  like 
intensity  to  those  which  reach  the  point  in  question,  but  differ 
from  them  by  a  half  wavelength,  or  at  least  by  an  odd  num- 


CORONAS  131 

ber  of  half  wavelengths,  in  phase.  Thus,  if  the  interchange 
between  the  opaque  and  the  transparent  parts  of  the  screen 
is  made,  the  intensity  of  illumination  at  these  points  remains 
unchanged.  Although  this  is  but  a  special  case,  the  char- 
acter of  the  reasoning  is  perfectly  general,  and  we  are  there- 
fore led  to  accept  the  principle  as  completely  established  and 
applicable  to  all  cases. 

By  means  of  the  principle  of  Babinet  we  pass  at  once  from 
the  case  of  the  screen  irregularly  pierced  with  uniform  circu- 
lar air  holes  to  the  glass  plate  covered  with  irregularly  dispersed 
disks  or  spheres  of  uniform  size,  and  finally,  to  small  spheres 
of  water  suspended  in  the  atmosphere.  That  the  spherules 
of  water  are  essentially  opaque  follows  at  once  from  the  fact 
that  only  an  infinitesimal  part  of  the  light  which  passes 
through  them  can  reach  the  eye,  on  account  of  their  short 
focal  length.  The  necessary  conditions  for  the  existence  of 
a  corona  are,  therefore,  suspended  particles  of  water  of  very 
uniform  size,  so  small  that  the  diffraction  rings  produced  by 
them  shall  be  considerably  larger  in  angular  dimensions  than 
the  sun  or  moon.  As  the  spherules  grow  larger  the  corona 
becomes  smaller,  and  vice  versa  ;  thus  we  have  in  the  coronas 
a  guide  as  to  whether  precipitated  moisture  is  increasing  or 
diminishing,  and  an  indication  of  value  in  predicting  the 
changes  of  weather. 

There  are  other  cases  of  phenomena  of  the  kind  under  dis- 
cussion and  bearing  a  similar  interpretation.  Most  observers 
would  probably  recognize  a  system  of  colored  rings  surround- 
ing an  electric  arc -light  seen  against  a  dark  background, 
especially  immediately  after  sleep.  Often  after  an  eye  has 
suffered  some  injury  —  as  from  a  blow,  for  example  —  these 
rings  appear  with  unaccustomed  brightness,  and  remain  for  a 
long  time  with  decreasing  intensity  and  concurrent  increas- 
ing dimensions,  until  they  resume  their  normal  appearance. 
They  are  attributed  to  slight  opacity  in  the  epithelial  cells 
of  the  cornea.  Occasionally  one  may  observe  such  circles 
about  a  light  when  seen  through  a  sheet  of  glass  upon  which 
there  is  a  considerable  deposit  of  moisture  from  the  air, 


132 


LIGHT 


although  this  indicates  a  regularity  of  the  deposit,  which  is 
rather  unusual. 

Rainbows  are  produced  by  refraction  and  reflection  of 
direct  sunlight  falling  upon  spherical  drops  of  water.  Ordi- 
narily very  little  light  reaches  the  eye  from  such  an  illumi- 
nated drop,  for  that  which  leaves  the  drop,  either  after  simple 
reflection  from  the  first  surface  or  after  more  complicated 
refractions  and  reflections  involving  the  rear  portion,  will 
be  so  widely  scattered  that  a  very  minute  fraction  of  the 
whole  can  fall  on  the  small  area  of  the  pupil.  There  are 
certain  definite  directions,  however,  where  this  conclusion 


FIGURE  34. 

does  not  hold.  The  accompanying  Figure  34  and  Figure 
35  will  help  to  make  this  statement  clear.  In  the  former 
let  o  be  the  centre  of  a  drop  of  water,  and  AB  a  plane 
wave-surface  moving  from  left  to  right  and  ultimately  fall- 
ing upon  it;  in  entering  the  sphere  every  portion  will  have 
its  curvature  modified  and  its  direction  of  propagation 
altered.  It  is  not  difficult  to  recognize  that  these  modifi- 
cations will  increase  with  the  distance  of  the  portion  of  the 
wave-surface  from  the  vertex  v.  Admitting  the  validity  of 
this  conclusion,  there  must  be  a  portion  of  the  wave -surf  ace, 
such  as  that  represented  at  a,  for  example,  in  which  the 
curvature  produced  by  refraction  will  be  just  sufficient  to 


RAINBOWS  133 

make  its  centre  fall  exactly  on  a  point  at  the  rear  surface 
of  the  drop,  whence  a  portion  suffers  reflection  toward  the 
front  of  the  drop.  It  is  this  reflected  portion  alone  which 
further  concerns  us.  This,  as  follows  at  once  from  relations 
of  symmetry,  will  reach  a  point  6,  at  the  same  distance  from 
the  point  at  which  reflection  takes  place  —  c  in  the  figure  —  as 
is  a ;  here  it  is  submitted  to  a  second  refraction,  which  just 
restores  the  portion  of  the  wave-surface  to  a  plane.  From 
this  point  the  light  will  be  propagated  without  further  change 
of  shape,  as  far  as  the  vertical  direction  is  concerned ;  and  to 
an  eye  placed  anywhere  in  its  course  the  drop  will  appear  to 
be  a  bright  object.  Light  which  falls  on  almost  any  other 
point  of  the  drop  will  be  so  spread  out  after  reflection  that  it 
will  add  very  little  to  that  due  to  the  former  source.  Calcu- 
lation shows  that  for  any  transparent  sphere  of  a  substance 
possessing  the  optical  constants  of  water  the  points  #,  e,  5, 
are  so  related  that  the  total  change  of  direction  of  motion 
of  the  light  is  nearly  138°.  Generalizing  from  these  facts, 
we  are  led  to  the  statement  that  every  drop  upon  which  the 
sun  is  shining  and  which  happens  to  be  138°  from  the 
sun  will  appear  to  emit  light,  and  as  all  such  drops  lie  in 
a  cone  whose  apex  is  at  the  eye,  the  result  will  be  a  circle 
of  light  at  the  angular  distance  given,  or,  what  amounts  to 
exactly  the  same  thing,  a  circle  having  its  centre  directly 
opposite  and  with  a  radius  of  42°.  An  obvious  consequence 
of  the  theory  is  that  no  rainbow  of  this  kind  can  be  seen 
when  the  sun  is  more  than  42°  above  the  horizon. 

Before-  passing  to  the  explanation  of  another  kind  of  rain- 
bow, a  remark  on  the  brightness  of  the  bow  may  be  made. 
Although  it  has  been  pointed  out  that  the  light  which  forms 
the  rainbow  is  propagated  without  loss  in  the  meridian  plane, 
on  account  of  its  flatness  in  that  direction,  there  is  no  such 
conservation  in  a  plane  perpendicular  to  this.  Hence  one 
might  conclude  that  the  intensity  of  illumination  in  a  bow 
would  rapidly  fall  off  with  increasing  distance  of  the  drops 
which  produce  it,  nor  would  there  be  any  escape  from  this 
conclusion  were  there  not  the  compensatory  fact  that  with 


134  LIGHT 

increasing  distance  the  number  of  drops  which  are  near 
enough  to  the  required  direction  to  take  part  in  the  phe- 
nomena would  increase  in  nearly  the  same  ratio. 

The  case  considered  is  not  the  only  one  in  which  a  plane 
incident  wave-surface  may  emerge  as  a  plane  in  an  altered 
direction,  as  will  appear  from  Figure  35.  Here,  at  a  larger 
distance  from  the  vertex  t>,  a  greater  deviation  and  change 
of  curvature  is  produced  by  refraction,  so  that  the  wave- 
surface  is  concave  until  it  reaches  a  point  cl9  after  which  it  is 
convex  until  incident  on  the  surface  of  the  sphere  where  it 
suffers  another  change  of  curvature  by  a  partial  reflection. 
If  the  proper  geometrical  conditions  are  met,  this  second 


35. 


change  in  curvature  —  always  in  the  direction  of  a  diminished 
value,  since  the  surface  is  concave  —  may  be  just  sufficient  to 
restore  the  wave  to  its  original  state  of  flatness,  in  which 
case  it  will  follow  the  course  indicated  by  the  dotted  line 
and,  after  experiencing  a  reflection  at  d^  which  would  again 
reduce  the  curvature  by  a  like  amount  so  that  the  centre  is 
at  <?2,  will  attain  the  surface  at  6,  where  a  final  modification 
would  restore  the  original  flatness.  The  number  of  different 
ways  in  which  this  same  general  result  may  be  accomplished 
is  boundless,  the  geometrical  conditions  being  that,  if  there 
is  an  odd  number  of  interior  reflections,  the  middle  one 
must  correspond  to  a  point  where  the  curvature  of  the  wave- 
surface  is  infinite  ;  while,  if  the  number  of  interior  reflections 


RAINBOWS  135 

is  even,  the  middle  chord  which  marks  the  path  of  the  light 
corresponds  to  a  region  where  the  curvature  is  zero.  The 
two  preceding  figures  are  constructed  to  represent  the  cases  of 
one  and  two  interior  reflections  for  water.  It  will  be  observed 
that  in  the  first  case  the  direction  of  the  emergent  wave  is 
changed  more  than  a  right  angle  from  its  original  direction, 
but  less  than  two  right  angles ;  in  the  second  case  the  deflec- 
tion is  between  two  and  three  right  angles.  Calculation 
yields  the  following  values  of  the  deflections  for  a  number 
of  reflections  varying  from  one  to  five:  — 

No.  of 
Reflections.  Deviations. 

1 138° 

2  .     .     .     . 231° 

3 318° 

4 404° 

5 486° 

These  are  given  to  the  nearest  degree  for  red  light. 

From  this  table  we  may  conclude  that  there  is  a  second 
locus  of  drops  which  send  light  to  the  eye,  namely,  a  circle 
everywhere  231°  from  the  sun,  or,  what  is  exactly  equiv- 
alent, at  an  angular  distance  of  —  51°  from  the  point  oppo- 
site to  the  sun.  The  minus  sign  attaching  to  the  last 
number  means  that  the  light  which  comes  to  us  from  the 
second  bow  has  entered  the  drops  on  the  lower  side,  instead 
of  on  the  upper  as  in  the  first  case.  The  second  rain- 
bow is  necessarily  much  fainter  than  the  first,  or  inner  one, 
since  its  light  has  experienced  two  partial  reflections  instead 
of  one ;  still,  it  is  always  visible  when  the  primary  is  very 
bright. 

What  is  true  relative  to  the  faintness  of  the  secondary  bow 
is  true  in  a  far  higher  degree  of  those  bows  of  a  higher  order, 
the  existence  of  which  is  demonstrated  by  the  foregoing  table, 
and  this  is  quite  sufficient  to  account  for  the  fifth  bow  never 
having  been  detected ;  but  were  the  third,  and  perhaps  even 
the  fourth,  as  favorably  situated  for  observation  as  are  the 


136  LIGHT 

primary  and  secondary,  it  would  certainly  be  visible  on 
occasions.  When  one  notes,  however,  that  both  these  are 
little  more  than  40°  from  the  sun,  and  that  this  portion  of 
the  sky  must  be  very  strongly  illuminated  by  the  light  which 
has  passed  through  the  drops,  it  is  easy  to  see  that  the  con- 
ditions are  most  unfavorable. 

This  explanation  of  the  rainbow  we  owe  in  nearly  the 
same  form  to  Descartes;  but  as  he  understood  nothing  of 
the  laws  of  dispersion,  it  was  left  for  his  great  follower, 
Sir  Isaac  Newton,  to  bring  the  theory  of  the  rainbow  nearly 
to  completion.  The  color  of  the  bow  is  obviously  due  to  the 
differing  refrangibility  of  different  wavelengths,  and  it  is 
quite  easy  to  extend  the  foregoing  reasoning  to  include  the 
consequences  of  this  fact.  For  example,  in  the  figures 
given  to  illustrate  the  paths  of  the  light  in  raindrops, 
and  which  are  drawn  for  red  light,  it  is  evident  that  since 
the  change  of  direction  by  refraction  is  always  greater  for 
short  than  for  long  waves,  in  the  first  one,  the  deflection 
for  blue  light  in  the  primary  bow  would  be  more  than  138° 
(almost  140°  in  fact)  and  its  radius  would  be  less  than 
42°  ;  hence  the  primary  bow  would  have  the  prismatic  series 
of  colors,  beginning  with  red  on  the  outside  and  termi- 
nating with  violet  on  the  inside.  For  the  secondary  bow, 
in  the  same  way,  we  would  have  a  deviation  greater  than 
231°  (about  234°)  for  the  blue,  and  consequently  a  circle 
with  a  radius  of  more  than  51° ;  hence  the  secondary  bow 
is  red  on  the  inner  side  and  violet  on  the  outer.  With  the 
added  statement  that  the  prismatic  colors  cannot  be  very 
pure  on  account  of  the  considerable  angular  diameter  of  the 
sun,  we  have  explained,  with  one  exception,  all  the  promi- 
nent and  familiar  features  of  the  rainbow.  Before  describ- 
ing this  and  certain  other  less  striking  phenomena,  it  will 
be  necessary  to  consider  another  point  which  the  diagrams 
teach. 

In  certain  determinate  directions  the  intensity  of  the  light 
leaving  the  drops  is  much  greater  than  in  all  others;  but 
this  is  equivalent  to  saying  that  near  certain  points  all  por- 


RAINBOWS  137 

tions  of  the  incident  wave  are  deflected  by  nearly  the  same 
amount,  or,  in  other  words,  that  the  angle  of  deviation 
changes  very  slowly  with  the  angle  of  incidence.  This,  how- 
ever, is  the  characteristic  of  maximum  and  minimum  values 
of  a  variable,  as,  for  instance,  when  the  sun  is  changing  its 
distance  from  the  horizon  most  slowly  it  is  either  at  its 
highest  or  lowest  point;  or  when  the  days  are  changing  in 
length  most  slowly,  they  are  either  at  their  longest  or 
shortest.  In  the  case  under  consideration  the  deviations 
cannot  be  maximum  values  in  either  the  one  or  the  other,  for 
a  single  reflection  of  that  portion  of  the  wave  which  passes 
through  the  centre  of  the  sphere  would  give  a  deviation  of 
180°,  and  after  two  reflections  of  360°.  These  values  being 
respectively  greater  than  138°  and  231°,  we  conclude  that 
the  particular  deviations  which  are  efficacious  in  producing 
the  bows  are  minimum  values.  From  this  we  may  make  the 
highly  important  deduction  that  no  drops  nearer  to  the  sun 
than  138°  can  send  any  light  to  the  eye  by  means  of  a 
single  reflection,  and  none  nearer  than  231°  can  send  any 
light  by  means  of  two  interior  reflections ;  consequently  those 
drops  which  are  situated  between  the  two  bows,  being  nearer 
in  one  direction  than  the  first  limit  and  in  the  other  than 
the  second,  can  send  no  light  either  by  a  single  or  a  double 
interior  reflection.  This  is  the  explanation  of  the  relative 
darkness  of  the  sky  between  the  bows,  which  forms  one  of 
the  most  conspicuous  features  of  a  well-developed  rainbow. 

Since  the  light  which  produces  the  bows  comes  from  por- 
tions of  the  plane  wave-surfaces  which  have  suffered  a  mini- 
mum deviation,  it  follows  that  there  are  portions  of  these 
surfaces  above  and  below  the  most  efficacious  part  which 
have  the  same  deviation,  such  light  reaching  the  eye  from 
drops  just  inside  the  primary  and  outside  the  secondary  bow. 
But  such  portions  will  have  passed  through  slightly  different 
lengths  of  water,  and  thus  have  experienced  slightly  differ- 
ent retardations.  Such  conditions  give  rise  to  the  phenomena 
of  interference.  It  is  not  difficult  to  see  that  this  will  pro- 
duce, inside  the  primary  and  outside  the  secondary,  a  series  of 


138  LIGHT 

repetitions  of  each  color,  of  rapidly  diminishing  brightness, 
and  at  an  angular  distance  decreasing  with  increasing  size  of 
the  drops.  Repetitions  of  this  character  are  called  super- 
numerary bows,  and  they  are  not  infrequently  seen  under 
the  highest  part  of  a  bright  inner  bow  and  more  rarely  above 
the  vertex  of  the  secondary  bow.  The  conditions  of  distinct- 
ness are  smallness  and  uniformity  in  size  of  the  drops,  which 
conditions  are  more  likely  to  be  found  well  up  in  the  air. 
Another  effect  of  interference  is  slightly  to  change  the  dimen- 
sions of  the  bows,  decreasing  the  apparent  diameter  of  the 
inner  and  increasing  that  of  the  outer.  The  changes,  al- 
though small  for  ordinary  rainbows,  'may  amount  to  several 
degrees  for  the  white  bow,  which  will  engage  our  attention 
next. 

A  lunar  rainbow  appears  almost  colorless  on  account  of  its 
faintness,  just  as  foliage  loses  nearly  every  trace  of  color  by 
moonlight,  but  this  is  a  merely  physiological  effect.  Occa- 
sionally, however,  a  very  bright  primary  bow  is  seen  with 
only  a  tinge  of  red  on  the  outer  and  of  blue  on  the  inner 
side.  This  occurs  when  a  bright  sun  shines  on  a  dense  wall 
of  fog  where  the  droplets  are  sufficiently  large  to  give  a  toler- 
ably definite  reflection,  but  at  the  same  time  differ  greatly 
among  themselves  in  size.  In  accordance  with  the  fact  stated 
in  the  preceding  paragraph  such  a  bow  is  always  smaller  in 
angular  diameter  than  a  colored  bow. 

Sometimes  arcs  of  rainbows  are  seen  on  a  sward  when 
covered  with  dew,  and  also  in  the  spray  thrown  up  by  the 
bow  of  a  vessel.  In  such  cases,  although  the  images  formed 
on  the  retina  are  always  circular  arcs,  we  are  apt  to  ascribe 
to  them  the  form  of  their  projections  upon  the  surface  below ; 
hence,  if  the  sun  has  an  altitude  of  more  than  42°,  they 
appear  to  be  arcs  of  ellipses ;  if  less  than  this,  arcs  of 
hyperbolas. 

The  theory  of  the  supernumerary  bows,  which  completed 
the  work  of  Descartes  and  of  Newton,  was  first  suggested  by 
Young  in  1804  and  completely  worked  out  by  Airy  in  1836. 

The  phenomena  caused  by  crystals  of  ice  suspended  in  the 


HALOS  139 

air  are  much  more  complicated  than  those  due  to  spheres  of 
water,  and  some  of  them  are  of  even  more  frequent  occur- 
rence, at  least  in  high  latitudes.  All  these  phenomena,  com- 
plex as  they  are,  may  be  divided  into  two  classes,  namely, 
those  produced  by  crystals  whose  axes  are  directed  purely 
fortuitously,  and  those  due  to  crystals  which  for  some  reason 
have  their  axes  parallel  to  a  fixed  line  or  to  a  fixed  plane. 
The  former  of  these  two  classes  may  be  explained  in  a  few 
words,  since  the  theory  is  very  simple. 

Crystals  of  ice  assume  a  very  large  variety  of  forms,  as  is 
well  known  to  every  one  who  has  observed  the  shapes  of  snow- 
flakes,  but  they  are  all  built  upon  a  general  plan  which  has 
the  right  hexagonal  prism  for  its  foundation.  It  will  be 
necessary,  however,  to  assume  the  presence  of  the  simplest 
form  only  in  the  following  discussion,  that  is,  of  simple  hex- 
agonal prisms ;  but  of  these  we  shall  suppose  that  there  are 


B 


FIGURE  36. 

two  types  such  as  are  represented  in  Figure  36,  where  A  is  a 
flat  hexagonal  plate  and  B  a  rod-like  or  elongated  prism  with 
a  hexagonal  base.  Both  these  forms  have  been  repeatedly 
observed.  If  one  of  these  transparent  bodies  is  considered 
as  an  optical  apparatus,  we  perceive  that  it  presents  six  sixty- 
degree  prisms  (in  the  sense  in  which  this  term  is  used  in 
optics),  and  twelve  right-angled  prisms,  the  former  kind 
being  limited  by  a  lateral  face  and  its  alternate  and  the  latter 
kind  by  a  lateral  face  and  either  base.  If  flat  wave-surfaces 
fall  upon  any  one  of  these  eight  faces,  provided  that  the  angle 
of  incidence  lies  within  a  certain  determinable  range  of  values, 
it  will  emerge  from  the  other  face  of  the  prism  deviated  by 


140  LIGHT 

an  amount  which  admits  of  ready  calculation,  since  it  de- 
pends upon  the  angle  and  upon  the  refractive  power  of  ice. 
Thus  for  a  sixty-degree  prism  we  find  a  minimum  deviation 
of  22°,  corresponding  to  an  angle  of  incidence  at  the  first  sur- 
face of  about  41°,  since  the  index  of  refraction  of  ice  is  1.31. 
If  this  angle  of  incidence  becomes  either  greater  or  less,  the 
proportion  of  light  transmitted  —  nearly  total  in  the  first  case 
—  very  rapidly  diminishes.  If,  therefore,  there  should  happen 
to  be  such  a  crystal  of  ice  suspended  in  the  air  at  an  angular 
distance  of  22°  from  the  direction  of  the  sun,  provided  that  it 
happened  to  have  its  face  in  the  proper  direction,  it  would 
send  all  the  light  which  enters  at  one  face  toward  the  eye  of 
the  observer.  At  a  very  small  distance  further  from  the  sun 
the  crystals  might  send  a  much  diminished  quantity  of  light 
to  the  eye,  but  at  a  still  less  distance  none  whatever,  since 
22°  is  the  minimum  deflection.  If  a  host  of  such  crystals 
were  uniformly  distributed  through  the  air,  it  is  obvious  that 
we  should  have  a  circle  of  light  44°  in  diameter,  with  the  sun 
at  its  centre.  When  we  take  into  consideration  the  secondary 
phenomenon  of  dispersion,  we  conclude  that  this  ring  must 
have  a  red  inner  border  and  a  much  less  distinct  bluish  outer 
border.  This  constitutes  the  familiar  halo  about  the  sun 
or  moon  which  may  be  seen  perhaps  as  often  as  fifty  to  a 
hundred  days  a  year  in  our  latitudes.  The  explanation  of 
the  phenomenon  is  due  to  Mariotte. 

The  rectangular  prisms  which  are  formed  by  the  dihedral 
angles  at  the  bases  of  the  crystals  give  rise  in  a  strictly 
similar  way  to  a  ring  with  a  radius  of  44°,  since  the  low 
refractive  power  of  ice  permits  light  to  pass  through  a  prism 
even  as  obtuse  as  this.1  On  account  of  the  far  more  critical 
arrangement  required  for  a  ninety-degree  prism  to  be  effective, 
as  well  as  on  account  of  the  much  greater  area  of  the  sky 
occupied  by  this  second  halo,  it  is  very  rarely  seen  in  its 
entirety ;  but  certain  portions  of  it,  as  will  appear  later,  are 
often  conspicuous.  The  first  suggestion  of  this  explanation 
is  attributed  to  Cavendish.  These  two  concentric  rings  are 

1  See,  however,  remarks  in  Appendix  C  with  regard  to  this  point. 


HALOS  141 

the  only  features  which  are  due  to  ice  crystals  whose  axes 
have  purely  fortuitous  directions. 

It  is  worth  noting  that,  as  no  cryst'als  nearer  the  sun  than 
22°  can  send  light  to  the  eye  by  refraction,  the  portion  of 
the  sky  immediately  inside  the  inner  ring  is  darker  than  the 
region  exterior  to  it.  This  is  a  very  conspicuous  feature  in 
a  well-developed  halo. 

Hexagonal  plates  of  the  shape  of  A  in  Figure  36,  and  elon- 
gated prisms  like  those  represented  at  B  in  the  same  figure, 
would  not,  in  general,  fall  through  the  air  with  their  major 
dimensions  indifferently  directed  with  respect  to  their  line  of 
motion.  If  the  air  were  quite  calm  the  plates  would  fall 
with  their  axes  vertical,  that  is  to  say,  the  plates  themselves 
would  be  horizontal,  while  the  second  form  would  fall  with 
their  axes  horizontal.  Of  the  latter  type  we  may  even  add 
that  the  prevailing  arrangement  may  be  that  of  a  maximum 
section  remaining  continuously  horizontal,  and  with  it,  of 
course,  two  of  the  lateral  faces  of  the  prisms,  although  in 
many  cases  —  doubtless  the  majority  of  cases  —  we  must 
imagine  the  crystals  constantly  rotating  on  their  horizontal 
axes  in  their  downward  motion.  The  reasons  for  these 
definite  arrangements  are  readily  explicable,  though  we  need 
not  stop  to  give  them  here.  It  is  sufficient  to  recall  the  fact 
that  such  a  body,  as  a  match  for  example,  cannot  be  made  to 
fall  through  any  considerable  distance  endwise,  nor  can  a 
small  flat  body,  like  the  petal  of  a  flower,  for  example,  be 
made  to  fall  edgewise. 

The  phenomena  resulting  from  the  refraction  of  light  by 
such  directed  crystals  must  obviously  bear  some  relation  to 
the  directions  involved,  which  are  only  that  of  the  sun  and 
the  vertical  direction,  or  that  of  the  zenith;  hence  all  the 
resulting  forms  must  be  symmetrical  with  respect  to  a  plane 
through  these  two  points  on  the  celestial  sphere.  This 
geometrical  consideration  will  lighten  the  description  of  the 
more  complex  features  of  halos. 

As  the  vertical  height  of  the  sun  above  the  horizon  must 
have  a  material  influence  on  the  aspect  of  the  halo,  we  are 


142 


LIGHT 


obliged  to  consider  the  problems  presented  when  this  element 
is  varied,  in  order  to  secure  a  quite  definite  notion  of  what  may 
be  seen.  In  turning  our  attention  first  to  the  halo  of  a  low 
sun,  we  shall  meet  not  only  with  the  features  which  are  most 
frequently  recorded,  but  we  may  gain  much  by  a  preliminary 
study  of  some  well-described  halo  of  complete  development. 
None  is  better  suited  to  our  purpose  than  a  very  extraordi- 
nary one  seen  by  Parry  and  Sabine  in  1820,  at  their  winter 
quarters  at  Melville  Island,  and  described  by  them  with  a 
scientific  precision  hardly  found  elsewhere.  Figure  37  is  a 


_ ...u 


FIGURE  37. 

faithful  copy  of  their  drawing,  and  it  is  followed  by  a  tran- 
script of  the  accompanying  description : l  — 

"  From  half-past  six  till  eight  A.M.,  on  the  9th,  a  halo,  with 
parhelia,  was  observed  about  the  sun,  similar  in  every  respect  to 
those  described  on  the  5th.  At  one  P.M.  these  phenomena  re- 
appeared, together  with  several  others  of  the  same  nature,  which, 
with  Captain  Sabine's  assistance,  I  have  endeavored  to  delineate 
in  the  annexed  figure. 

s,  the  sun,  its  altitude  being  about  23°.     h,  h,  the  horizon. 

l  Parry's  First  Voyage,  pp.  164-165. 


HALOS  143 

t,  Uj  a  complete  horizontal  circle  of  white  light  passing  through 
the  sun. 

a,  a  very  bright  and  dazzling  parhelion,  not  prismatic. 

b,  c,  prismatic  parhelia  at  the  intersection  of  a  circle  a,  bt  d,  c, 
whose  radius  was  22£°  with  the  horizontal  circle  t,  u. 

x,  d,  v,  an  arch  of  an  inverted  circle,  having  its  centre  appar- 
ently about  the  zenith.  This  arch  was  very  strongly  tinted  with 
the  prismatic  colors. 

k,  e,  I,  an  arch  apparently  elliptical  rather  than  circular,  e  being 
distant  from  the  sun  26°  ;  the  part  included  between  x  and  v  was 
prismatic,  the  rest  white.  The  space  included  between  the  two 
prismatic  arches,  xevd,  was  made  extremely  brilliant  by  the  re- 
flection of  the  sun's  rays,  from  innumerable  minute  spiculae  of 
snow  floating  in  the  atmosphere. 

qfr,  a  circle  having  a  radius  from  the  sun,  of  45°,  strongly  pris- 
matic about  the  points  fqr,  and  faintly  so  all  round. 

mn,  a  small  arch  of  an  inverted  circle,  strongly  prismatic,  and 
having  its  centre  apparently  in  the  zenith. 

rp,  qo,  arches  of  large  circles,  very  strongly  prismatic,  which 
could  only  be  traced  to  p  and  o ;  but  on  that  part  of  the  hori- 
zontal circle  tu,  which  was  directly  opposite  to  the  sun,  there 
appeared  a  confused  white  light,  which  had  occasionally  the 
appearance  of  being  caused  by  the  intersection  of  large  arches 
coinciding  with  a  prolongation  of  rp  and  qo. 

The  above  phenomena  continued  during  the  greater  part  of  the 
afternoon;  but  at  six  P.M.,  the  distance  between  d  and  e  in 
creased  considerably,  and  what  before  appeared  an  arch,  x,  d,  v, 
now  assumed  the  appearance  given  in  fig.  12,  plate  287,  of 
Brewster's  Encyclopaedia,  resembling  horns,  and  so  described  in 
the  article  '  Halo/  of  that  work.  At  90°  from  the  sun,  on  each 
side  of  it,  and  at  an  altitude  of  30°  to  50°,  there  now  appeared 
also  a  very  faint  arch  of  white  light,  which  sometimes  seemed  to 
form  a  part  of  the  circles  qo,  rp  ;  and  sometimes  we  thought 
they  turned  the  opposite  way.  In  the  outer  large  circle  we  now 
observed  two  opposite  and  corresponding  spots  ?/,  y,  more  strongly 
prismatic  than  the  rest,  and  the  inverted  arch  m,  /,  n,  was  now 
much  longer  than  before,  and  resembled  a  beautiful  rainbow." 

In  order  to  make  clear  the  following  discussion  of  the 
phenomena  due  to  directed  crystals,  we  must  find  a  means  of 


144  LIGHT 

distinguishing  the  different  faces.  This  may  readily  be  done 
by  reference  to  the  accompanying  Figure  38,  where  p0  to  jt?5, 
inclusive,  represent  the  six  lateral  faces 
of  the  hexagonal  prisms,  alternate  faces 
having  either  even  or  odd  suffixes  ;  o 
represents  a  base,  and  if  we  wish  to 
indicate  the  opposite  base  which  is  the 
only  remaining  face  of  the  prism,  we 
shall  designate  it  by  o'. 

It  is  not  necessary  to  dwell  upon  the 
FIGURE  38.  crystals  of  the  first  group,  that  is,  those 

which  have  a  purely  fortuitous  arrange- 
ment of  their  axes  and  the  presence  of  which  is  clearly 
indicated  by  the  concentric  circles  acdb  and  qfr,  since 
these  phenomena  have  been  already  considered ;  but  we  may 
turn  at  once  to  the  more  complicated  features.  As  the 
simpler  of  the  others,  we  will  review  the  effects  due  to  the 
second  group,  that  is,  to  plate  crystals  figured  in  A^  Figure  36, 
which  fall  with  their  axes  perpendicular.  Of  all  such  crystals 
which  happen  to  be  at  a  distance  of  22°  to  the  right  or 
left  of  the  sun,  a  very  great  number  would  be  found  at  a 
given  instant  in  such  a  position  that  light  entering  one  verti- 
cal lateral  face  would  emerge  from  an  alternate  face  with  a 
total  deviation  of  22°.  These  produce  the  prismatic  par- 
helia c  and  b.  Crystals  of  the  same  kind  slightly  more  dis- 
tant from  the  sun  might  send  light  to  the  observer  in  two 
different  ways,  namely,  either  by  refraction  at  an  angle  differ- 
ing from  that  of  minimum  deviation,  or  by  refraction  at  the 
entering  face  followed  by  total  reflection  on  the  adjacent  face 
and  a  final  refraction  at  the  face  of  emergence.  In  the  for- 
mer case  light  of  less  refrangibility  would  be  superimposed 
upon  that  from  prisms  more  nearly  oriented  to  the  position 
of  minimum  deviation ;  consequently  light  from  these  direc- 
tions would  tend  toward  colorlessness.  In  the  latter  case  the 
light  would  be  pure  white ;  since  the  dispersion  produced  at 
one  refraction  would  be  compensated  at  the  other.  From 
these  considerations  we  see  that  if  this  group  of  crystals  were 


HALOS  145 

very  well  represented  the  parhelia  ought  to  have  a  vivid 
prismatic  red  color  on  the  side  toward  the  sun  and  a  bright 
colorless  tail  stretching  out  for  a  number  of  degrees  in  the 
opposite  direction.  Since  the  last-named  feature  was  not 
noted  by  the  observers,  although  it  is  most  conspicuous  in  a 
large  number  of  recorded  halos,  we  must  conclude  that  the 
group  A  was  not  very  well  represented  in  the  case  under 
discussion.  Each  of  the  vertical  faces  of  these  crystals 
would  somewhere  produce  by  reflection  an  image  of  the  sun 
which  must  lie  at  a  height  above  the  horizon  equal  to  that  of 
the  sun  itself ;  hence  the  total  effect  would  be  a  colorless 
horizontal  circle  passing  through  the  sun  and  extending 
completely  around  the  heavens.  This  circle,  called  the  par- 
helic  circle,  is  represented  in  part  in  the  drawing  by  the  band 
tu.  Finally,  we  have  only  to  add  that  the  effect  produced 
by  the  rectangular  prisms,  one  side  of  which  in  this  case 
must  remain  constantly  horizontal,  is  a  colored  arc  mfn, 
which,  since  these  prisms  run  to  an  edge  and  are  not  inter- 
rupted by  an  intermediate  face  capable  of  adding  totally  re- 
flected light,  is  far  purer  in  its  colors  than  are  any  of  the 
other  portions  of  the  halo,  often  rivalling  the  brightest  rain- 
bow. The  reason  this  light  is  spread  into  an  arc  instead  of 
being  confined  to  a  bright  spot  similar  to  the  twenty-two- 
degree  parhelia  is  because  in  this  case  the  edges  of  the 
prisms  are  only  restricted  to  parallelism  to  a  horizontal  plane, 
not,  as  in  the  other  case,  to  parallelism  to  a  fixed  line.  It  is 
interesting  to  note  that  sometimes  this  arc  and  the  par- 
helia attributed  to  the  same  crystals  constitute  the  whole 
visible  phenomenon,  indicating  that  only  this  particular  group 
is  present. 

We  now  turn  to  the  consideration  of  the  effects  produced  by 
prisms  of  the  third  group  B.  We  shall  assume  that  some 
of  these  fall  with  their  maximum  sections  continuously  hori- 
zontal, in  which  case  we  have  the  two  bases  constantly 
vertical  and  two  lateral  faces,  which  for  convenience  we  will 
indicate  as  p0  and  p%,  constantly  horizontal.  Such  crystals 
as  happened  to  be  at  the  point  indicated  by  e  in  the  drawing, 

10 


146  LIGHT 

and  at  the  same  time  had  their  axes  nearly  at  right  angles 
to  the  vertical  plane  of  the  sun,  were  the  source  of  the  bril- 
liant prismatic  image  there  observed,  which  was  more  remote 
from  the  sun  than  d,  because  these  crystals  were  not  at  the 
position  of  minimum  deviation.  Crystals  of  the  same  kind 
to  the  right  and  left  of  these,  which  happened  to  have  for 
the  moment  a  proper  orientation  of  their  horizontal  axes, 
produced  similar  prismatic  images  at  a  somewhat  greater 
distance  from  the  sun  but  at  a  less  altitude,  whence  the  arc 
xev.  All  the  light  which  entered  a  lateral  face,  was  re- 
flected interiorly  from  a  base,  and  then  emerged  from  a  face 
parallel  to  the  first,  came  from  some  point  in  the  parhelic 
circle,  as  did  also  such  light  as  was  reflected  from  a  base 
without  entering  the  prism.  It  is  obvious  that  much  of  the 
light  which  suffered  this  modification  was  totally  reflected 
from  the  interior  and  would  yield  very  bright  images;  indeed, 
it  is  probable  that  in  this  particular  halo  almost  all  the 
parhelic  circle  is  to  be  attributed  to  this  group.  So,  too, 
most  of  the  light  in  the  arc  mfn,  produced  by  rectangular 
prisms  with  horizontal  edges,  therefore  also  identical  in 
action  with  the  corresponding  portion  of  the  crystals  of 
group  J.,  must  be  attributed  in  this  particular  halo  to  B 
crystals.  Finally,  the  arcs  qo  and  rp  of  the  drawing  were 
produced  by  the  action  of  this  group  by  means  of  the  rectan- 
gular prisms  which  were  underneath  arid  had  their  edges  at 
a  constant  angle  of  60°  with  the  horizon.  These  are  all 
the  features  pictured  in  the  drawing  which  owe  their  origin 
to  the  B  group,  although  we  shall  return  to  the  discussion 
of  another  one  mentioned  in  the  text,  which  under  certain 
circumstances  becomes  of  great  importance. 

The  stability  of  the  B  type  of  crystals  with  respect  to  the 
horizontality  of  the  maximum  cross-section  can  be  hardly 
regarded  as  very  pronounced  because  of  the  small  excess  of 
this  section  over  the  minimum,  which  is  but  30°  from 
it;  therefore  we  are  not  surprised  to  find  that  many  ice 
crystals  having  this  shape  do  not  fall  in  the  manner  indi- 
cated. Whatever  the  conduct  of  these  may  be,  unless,  in- 


HALOS  147 

deed,  they  simply  oscillate  about  the  position  of  stability, 
the  effect  produced  by  them  will  be  exactly  the  same  as 
though  they  rotated  at  a  constant  speed  around  their  hori- 
zontal axis.  To  specify  this  sub-type  of  columnar  crystals 
they  will  be  designated  by  the  symbol  B'.  Their  existence 
in  this  particular  halo  is  not  only  demonstrated  by  the  phe~ 
nomena  attributable  to  them  alone,  but  they  were  in  certain 
favorable  situations  clearly  seen  as  "  innumerable  minute 
spiculse  of  snow."  To  these  were  due  the  brighter  portions 
of  the  forty-six-degree  halo  in  the  neighborhood  of  its  tan- 
gent arcs,  the  arc  xdv  tangent  to  the  twenty-two-degree  halo, 
and  the  brilliant  white  light  above  the  last  of  which  the  color- 
less spot  on  the  horizon  below  the  sun  is  the  correlative.  The 
white  light  was  due  to  light  totally  reflected  from  the  interior 
of  crystals  which  happened  to  be  favorably  situated,  exactly 
in  the  same  way  as  explained  in  the  case  of  the  A  group  im- 
mediately outside  the  parhelia,  those  sufficiently  near  the  ob- 
server sparkling  in  the  sunshine.  The  anthelion,  a  bright  spot 
in  the  parhelic  circle  directly  opposite  the  sun,  was  also  pro- 
duced by  these  B'  prisms.  The  manner  of  production  may 
be  explained  as  follows :  Imagine  a  B'  crystal  anywhere  in  the 
line  from  the  observer  to  the  anthelion,  which  for  the  instant 
has  two  of  its  faces,  pQ  and  p$,  for  example,  perpendicular ; 
light  entering  one  of  these  faces  would  suffer  successive 
reflection,  in  either  order,  at  the  opposite  face  and  a  base,  and 
then  emerge  through  the  face  of  entry.  Such  light  would 
come  from  the  anthelion.  Now  imagine  the  crystal  to  rotate 
on  its  horizontal  axis;  it  will  obviously  recover  an  exactly 
equivalent  position  after  turning  60°,  but  in  the  mean 
time  the  image  formed  by  the  two  refractions  and  the  two 
intermediate  reflections  would  describe  an  oblique  path  pass- 
ing through  the  anthelion.  Since  this  reasoning  is  indepen- 
dent of  the  direction  of  the  horizontal  axes  with  respect  to 
the  vertical  circle  through  the  sun,  all  this  kind  of  crystals 
in  the  direction  indicated  would  give  rise  to  oblique  arcs 
having  the  point  horizontally  opposite  the  sun  in  common; 
hence  this  region  would  appear  brighter  than  the  surround- 


148  LIGHT 

ing  sky.  Since  they  lie  within  certain  limits  in  all  possible 
directions,  these  oblique  arcs  would  not  in  general  be  dis- 
tinguishable, but  with  a  low  sun  it  is  not  difficult  to  prove 
that  those  having  an  inclination  of  about  60°  with  the 
horizon  are  brighter  than  the  others,  and  such  arcs  seem  to 
have  been  recorded  a  considerable  number  of  times,  although 
they  must  be  classed  as  among  the  rarer  appendages  of  halos.1 
Now  consider  the  changes  recorded  by  Parry  and  Sabine 
as  the  sun  sank  toward  the  horizon.  First  of  all,  e  sepa- 
rated more  widely  from  d,  which  is  quite  in  accordance  with 
the  theory,  because  the  sun  is  still  further  removed  from  the 
position  of  minimum  deviation,  corresponding  to  an  altitude 
of  the  sun  of  about  49°,  than  in  the  case  represented 
by  the  drawing.  Then  the  points  indicated  by  yy  in  the 
outer  circle  began  to  grow  bright;  this  was  the  beginning  of 
a  phenomenon  complementary  to  the  tangent  arcs  qo  and  rp 
produced  by  a  refraction  through  a  lateral  face  in  momen- 
tarily favorable  position  and  a  base.  A  third  feature  de- 

1  As  we  shall  not  have  occasion  to  return  to  this  point,  and  as  it  has  given 
no  little  trouble  to  investigators,  it  may  be  worth  while  slightly  to  extend  the 
reasoning.  It  is  obvious  that  if  the  two  interior  reflections  were  both  partial  the 
emergent  light  would  be  extremely  faint  and  probably  quite  negligible;  therefore 
the  interesting  cases  are  when  one  of  these  reflections  is  total.  Since,  however, 
there  can  be  no  question  of  total  reflection  on  a  lateral  face,  because  light  which 
can  enter  such  a  face  can  evidently  emerge  from  the  opposite  one,  we  are 
confined  to  cases  of  total  reflection  on  a  base.  The  minimum  angle  of  incidence 
in  these  prisms  under  consideration  which  will  yield  such  iotal  reflection  is  57°.7  ; 
hence  only  those  whose  axes  have  a  direction  in  azmuth  differing  from  that  of 
the  sun  from  32°.3  to  90°  meet  the  condition  defined.  From  those  anywhere  near 
the  latter  limit,  however,  very  little  light  would  be  returned  on  account  of  the 
small  effective  area  of  the  base,  which  is  only  the  projected  area;  hence  much  the 
brightest  arcs  must  be  ascribed  to  those  having  an  angular  displacement  from 
the  sun's  vertical  not  less  nor  very  much  greater  than  32°.3.  With  a  higher  sun 
it  is  possible  to  have  light  other  than  that  which  has  entered  a  vertical  lateral 
face  fall  upon  the  interior  rectangular  mirror,  namely,  some  of  that  which  has 
entered  at  the  superior  adjacent  face  plt  and,  after  reflection,  emerged  at  p6.  In 
such  cases  it  would  be  possible  to  have  two  pairs  of  oblique  arcs  through  the 
anthelion,  which  also  seem  to  have  been  observed  on  rare  occasions.  Finally, 
we  may  add  that  an  anthelion  with  a  high  sun,  say  of  50°  or  more,  is  impossible, 
since  there  will  be  either  no  light  at  all  or  an  altogether  insignificant  amount 
incident  upon  the  two  faces. 


HALOS  149 

scribed  in  these  observations  is  of  especial  interest  because 
so  conspicuous  when  B  crystals  are  abundant  and  the  sun 
very  high,  namely,  the  faint  arcs  which  seem  to  stretch 
toward  the  anthelion  in  the  direction  of  a  prolongation  of 
the  tangent  arcs  to  the  lower  part  of  the  forty-six-degree 
halo.  These  were  produced  by  light  which  entered  the  B 
crystals  through  a  po  face  and,  after  an  intermediate  reflec- 
tion from  one  of  the  bases,  emerged  through  jt?2  or  p±.  Such 
arcs  would  be  very  faint  throughout,  but  brightest  at  a  dis- 
tance of  about  90°  from  the  sun. 

The  long  discussion  of  this  complicated  halo  which  we 
have  given  serves  two  different  purposes :  First,  it  gives  the 
explanation  of  a  considerable  part  of  all  the  recorded  phe- 
nomena of  halos;  second,  it  justifies  our  assumption  of 
the  three  types  of  crystals,  and  two  varieties  of  the  last  type. 
We  may  therefore  with  much  confidence  approach  the  expla- 
nations of  the  remaining  phenomena,  although  there  are 
perhaps  no  complex  cases  on  record  nearly  as  well  observed 
as  the  present  one. 

As  the  sun  attains  a  greater  altitude  above  the  horizon, 
a  larger  and  larger  portion  of  the  light  entering  an  upper 
horizontal  face  is  reflected  from  an  adjacent  vertical  face; 
hence,  in  general,  the  parhelic  circle  gains  in  brightness 
with  this  change,  especially  as  the  interior  reflections  be- 
come total  when  the  altitude  is  more  than  32°.  In  the 
presence  of  A  crystals  some  of  the  light  which  enters  an 
upper  base  may  suffer  successive  reflections  from  two  adja- 
cent sides  before  emerging  from  the  inferior  base.  This 
gives  rise  to  a  singular  phenomenon  not  infrequently  re- 
corded, which  is  known  under  the  name  of  the  paranthelion 
of  120°.  To  explain  it  a  well-known  law  of  optics  must 
be  stated.  If  an  image  of  an  object  is  formed  by  two  reflec- 
tions from  two  plane  mirrors,  its  angular  displacement  from 
the  object,  measured  in  a  plane  perpendicular  to  the  line 
of  intersection  of  the  mirrors,  is  exactly  twice  the  angle 
included  between  the  normals  to  the  mirrors.  The  phenom- 
enon discussed  in  connection  with  Figure  3,  page  9,  is  a 


150  LIGHT 

special  case  where  the  angle  between  the  mirrors  is  a  right 
angle.  In  the  case  in  question  the  angle  between  two 
adjacent  sides  is  60°;  hence  the  images  formed  of  the 
sun  would  lie  in  the  parhelic  circle  120°  from  the  sun  or 
60°  from  the  anthelion  on  either  side.  It  is  the  latter 
relation  that  gives  them  their  name.  Another  feature 
which  very  unusually  developed  and  abundant  A  crystals 
might  produce  is  a  pair  of  bright  spots  at  g  and  its  corre- 
sponding point  on  the  opposite  side  of  the  sun,  which  are 
called  the  parhelia  of  46°.  These  are  probably  second- 
ary images  of  the  inner  parhelia,  that  is  to  say,  images 
of  them  formed  by  refraction  exactly  in  the  same  way  as 
they  are  formed  with  the  sun  as  the  source  of  light. 

With  a  sun  much  higher,  say  half-way  toward  the  zenith, 
the  series  of  phenomena  will  be  very  considerably  modified. 
The  A  crystals  still  yield  their  parhelic  circle  with  its  par- 
helia well  outside  the  twenty-two-degree  circle,  also  the 
parhelia  of  120° ;  but  the  tangent  arc  of  the  forty-six-degree 
circle  can  no  longer  appear.  The  appendages  produced  by  the 
B  group  are  also  rather  inconspicuous,  although  they  add  to 
the  light  of  the  parhelic  circle  and  also  produce  short  oblique 
arcs  through  the  anthelion.  The  crystals  which  have  been 
designated  by  B'  may  still  give  rise  to  an  anthelion  as  before ; 
but  their  most  notable  contribution  to  all  the  phenomena 
is  an  oval  touching  the  twenty-two-degree  halo  at  its 
highest  and  lowest  points,  and  lying  wholly  outside  of  it. 
This  oval,  which  has  been  very  frequently  seen,  has  been 
perfectly  explained  by  Venturi,  who  showed  that  the  upper 
(kxdvl  of  Figure  37)  and  lower  tangent  arcs  of  the  twenty-two- 
degree  circle,  that  have  been  shown  as  attributable  to  the 
horizontal  crystals  without  fixed  direction  of  the  axial  sec- 
tions, bend  toward  each  other  with  increasing  altitude  of  the 
sun,  until  at  a  height  of  about  50°  they  unite  and  form  the 
oval.  It  is  easy  for  us  to  see  that  some  such  phenomenon 
must  follow,  if  we  reflect  that,  were  the  sun  at  the  zenith,  all 
the  B'  crystals,  like  the  undirected  crystals,  would  serve 
simply  to  produce  a  twenty-two-degree  halo ;  but  as  the  sun 


HALOS  151 

moves  from  the  zenith  those  crystals  which  are  in  the  same 
vertical  circle  would  alone  maintain  the  same  relation  to  it, 
those  not  in  this  vertical  transmitting  the  light  somewhat 
obliquely,  and  consequently  giving  a  greater  deviation  with 
a  resulting  locus  of  the  image  of  the  sun  at  a  greater  dis- 
tance than  22°.  Hence,  from  considerations  of  symmetry, 
we  recognize  that  this  locus  must  be  an  oval. 

At  a  still  higher  altitude  —  60°  or  more,  for  example  — 
the  A  crystals  are  of  little  significance  except  as  they  may 
add  to  the  effect  produced  by  the  rectangular  edges  of  the 
B  group  and  to  the  intensity  of  the  parhelic  circle.  Paran- 
thelia  are  invisible  or  inconspicuous  on  account  of  the  thin- 
ness of  these  prisms  and  the  large  angle  of  interior  incidence. 
On  the  other  hand,  the  B  group  of  crystals,  if  largely  repre- 
sented, becomes  very  important.  First,  they  may  produce  a 
strong  parhelic  circle;  second,  they  yield  from  light  which 
enters  a  base  and  emerges  at  a  lower  horizontal  face  a  tan- 
gent arc,  brightest  with  a  solar  altitude  of  68°,  to  the  lowest 
part  of  the  forty-six-degree  circle ;  finally,  from  light  which 
enters  at  p\  or  j95,  is  reflected  at  a  base,  and  emerges  at  the 
lower  horizontal  face,  they  produce  a  pair  of  peculiar  spiral 
curves,  which  intersect  at  the  anthelion  point,  but  fade  out 
before  reaching  a  second  crossing  of  the  vertical  circle  pass- 
ing through  the  sun.  Figure  39  represents  a  spherical 
projection  of  a  carefully  constructed  halo  of  this  type,  the 
constants  of  which  were  calculated  from  the  present  theory. 
The  zenith  distance  of  the  sun  is  25° ;  about  the  sun .  as 
a  (projected)  centre  are  drawn  the  unbroken  circles  repre- 
senting the  twenty-two-degree  halo,  while  with  the  zenith 
as  a  centre  and  a  radius  of  25°  the  double  circle  which 
represents  the  parhelic  circle  is  described  in  broken  lines. 
The  remaining  curve,  self  intersecting  at  /S",  gives  the  two 
spirals  under  discussion.  The  parts  drawn  in  smooth  lines 
are  the  brightest  portions;  most  of  the  remainder,  drawn 
in  broken  lines,  is  relatively  faint,  although  in  the  region 
immediately  below  the  sun  it  coincides  with  an  arc  which  is 
produced  by  incident  and  emergent  refractions  without  the 


152 


LIGHT 


intermediate    reflection.     Were   the   oval   of    Venturi    also 
present  we  should  here  find  three  distinct  arcs  which  are  the 


\ 


\ 


\ 


\ 


FIGURE  39. 

exact  complement  to  the  three  arcs  in  the  region  de  of  Parry 
and  Sabine's  drawing.1 

1  Halos  of  this  kind  are  practically  invisible  in  Europe,  on  account  of  the  high 
latitude  of  the  countries  where  alone  these  phenomena  are  seen,  but  in  the  United 
States,  where  the  climate  is  like  that  of  northern  Europe  and  the  latitude  that  of 
southern  Europe,  many  such  have  been  visible.  Sketches  of  extraordinary  halos 
of  this  type  may  be  found  by  the  curious  reader  in  the  "  American  Journal  of 
Sciences"  (1),  vols.  vii.,  x.,  xi.,  and  (2)  vol.  xl.,p.  394;  and  in  the  "Report  of  the 
British  Association,"  1861  (2),  p.  63.  None  of  these,  unfortunately,  is  very  critical, 
since  in  every  case  the  observer  seems  to  have  assumed  as  a  matter  of  course 
that  all  the  arcs  seen  must  be  parts  of  true  circles,  and  in  not  a  few  cases  he 
has  not  hesitated  to  complete  the  circles  in  accordance  with  his  own  conceptions. 
Nevertheless  the  fact  that  the  sole  feature  in  which  all  these  sketches  are 
in  agreement  is  that  derived  from  the  theory  developed  above,  and  pictured  in 
the  drawing,  may  be  taken  as  almost  conclusive  evidence  in  its  favor.  Further 
remarks  on  the  theories  of  halos  will  be  found  in  Appendix  C. 


HALOS  153 

There  remains  a  feature  of  undoubted  authenticity,  that 
is,  one  which  has  been  recorded  frequently  by  competent 
observers,  which  is  worthy  of  note,  namely,  vertical  columns 
of  light  occasionally  passing  through  the  sun  when  not  far 
from  the  horizon.  These  may  be  explained  as  follows: 
Since  the  larger  surfaces  of  the  A  group  of  crystals  are 
continuously  nearly  horizontal,  if  there  happen  to  be  many  of 
them  near  the  observer  and  between  him  and  the  sun,  they 
would  yield  a  specular  image  of  the  sun  which  would  appear 
above  or  below  that  body  according  as  it  happens  to  be  at 
the  moment  below  or  above  the  true  horizon.  If  the  crystals 
possess  a  slight  rocking  motion  in  their  fall  —  and  we  have 
abundant  evidence  that  this  sometimes  occurs,  not  only  by 
such  features  as  the  brightening  of  the  forty-six-degree  halo, 
as  explained  on  page  146,  but  also  by  the  arcs  known  as  the 
tangent  arcs  of  Lowitz,  which  extend  from  the  parhelia  ac- 
companying a  high  sun  downward  to  the  twenty-two-degree 
halo  —  the  specular  images  will  be  stretched  out  into  a  vertical 
column  in  much  the  same  way  as  an  image  of  the  sun  produced 
by  a  surface  of  quiet  water  is  altered  into  a  streak  of  light 
whose  axis  passes  through  the  sun  when  this  surface  is  ruffled 
by  ripples ;  and  this  quite  irrespective  of  the  direction  of  the 
motion  of  the  ripples.  It  is  evident  that  the  reflection  from 
the  lower  surfaces  of  the  prisms  in  question  will  be  total; 
hence  we  must  attribute  to  them  the  chief  role  in  this  phe- 
nomenon, and  very  little  to  the  faces  of  the  B  group,  some  of 
which  are  also  horizontal. 


CHAPTER  VIII 
THE  EYE   AND  VISION 

IN  its  general  structure  the  eye  is  a  camera  obscura,  but 
it  possesses  an  important  difference  from  the  common  camera 
in  that  the  interior  is  filled  with  media  not  unlike  water  in 
their  optical  properties.  This  peculiarity  carries  with  it  cer- 
tain interesting  modifications  in  the  phenomena  of  vision, 
which,  however,  can  hardly  be  more  than  barely  indicated 
here.  If  we  look  at  Figure  40,  which  represents  a  horizontal 
section  of  the  human  eye,  copied  after  Helmholtz's  careful 
drawing,  we  shall  be  enabled  to  understand  the  principal 
details.  The  whole  body  is  a  spheroid  somewhat  flattened 
in  the  direction  of  its  axis.  Within  is  found  an  anterior 
chamber  B,  which  is  filled  with  a  liquid  little  different  from 
pure  water,  and  hence  called  the  aqueous  humor;  and  the 
posterior  chamber  containing  a  somewhat  firm  transparent 
body  to  which  the  name  vitreous  humor  is  given.  The  latter 
is  marked  O  in  the  diagram.  Between  these  two  is  situated 
a  transparent  lenticular  body  A,  which  is  known  as  the 
lens.  The  front  of  B  is  bounded  by  a  very  transparent  con- 
vex skin  called  the  cornea,  which  projects  quite  perceptibly 
beyond  the  general  surface  of  the  eyeball.  Resting  upon 
the  surface  of  the  lens,  and  constituting  the  back  of  the 
anterior  chamber  B,  is  a  delicate  membrane  55,  having  a 
circular  hole  in  its  centre.  The  central  aperture  is  called 
the  pupil  of  the  eye,  and  the  membrane  which  it  pierces,  the 
iris.  It  is  the  latter  which  gives  the  characteristic  color  to 
the  eye,  and,  as  a  part  of  the  optical  apparatus,  its  office  is 
to  vary  conveniently  the  effective  diameter  of  the  lens.  The 


156 


LIGHT 


variation  in  the  size  of  the  pupil  is  brought  about  by  the 
contraction  of  one  or  the  other  of  two  sets  of  muscles, 
the  first  being  radially  disposed  so  that  contraction  enlarges 
the  aperture,  while  the  other,  arranged  in  a  circular  manner 
along  the  inner  edge  of  the  iris,  causes  a  diminution  of  the 
pupilar  opening  by  contraction.  These  muscles  respond  auto- 


matically  to  the  stimulus  of  light  on  the  eye,  so  that  when 
this  is  very  bright  the  pupil  becomes  extremely  small.  The 
ordinary  range  of  effective  diameter  of  the  pupil  may  be 
taken  as  from  somewhat  less  than  a  twentieth  of  an  inch  in 
bright  light  to  a  little  more  than  four  times  as  much  in  a 
very  faint  light,  thus  increasing  the  illumination  more  than 
sixteen  fold. 


THE  EYE  157 

The  cornea  and  lens  combine  as  an  optical  system  which 
alters  the  curvature  of  nearly  flat  wave-surfaces  falling  upon 
them,  so  that  after  modification  their  geometrical  centres 
fall  just  on  the  back  inner  surface  of  the  eye,  and  conse- 
quently form  there  an  extended  image  of  objects  sufficiently 
remote.  Over  this  rear  surface  is  spread  a  delicate  mem- 
brane i,  composed  chiefly  of  nerve  tissue  which  is  connected 
with  the  brain  by  a  bundle  of  nerve  fibres  passing  through 
the  orifice  d.  The  membrane  is  called  the  retina,  and  the 
bundle  of  nerves  the  optic  nerve.  The  aperture  for  the  optic 
nerve  lies  on  the  nasal  side  of  the  axis  of  the  eye ;  hence  the 
figure  represents  a  horizontal  section  through  the  right  eye 
when  seen  from  above.  Before  describing  more  in  detail  the 
construction  and  function  of  the  retina,  we  may  consider  the 
provision  for  securing  sharpness  of  definition  in  the  images 
on  the  retina,  when  the  distance  of  the  object  viewed  is 
changed.  If  the  cornea  and  lens  together  possess  just  suffi- 
cient power  to  cause  flat  wave-surfaces  to  have  their  centres 
on  the  retina,  they  would  prove  insufficient  for  convex  wave- 
surfaces  such  as  would  come  from  nearer  objects.  The 
means  taken  in  a  photographic  camera  to  adjust  for  such 
differences  is  to  alter  the  distance  between  the  lens  system 
and  the  screen  which  receives  the  images,  but  in  the  eye  this 
necessary  adjustment,  called  accommodation  in  this  case,  is 
wholly  different  and  without  analogy  in  any  artificial  opti- 
cal instrument.  In  short,  accommodation  is  produced  by  a 
change  in  the  absolute  power  of  the  lens  attending  a  varia- 
tion in  its  thickness.  Figure  41  clearly  shows  this  change, 
the  left  half  exhibiting  the  shape  of  the  lens  when  suited 
to  distinct  vision  of  a  distant  object,  and  the  right  half  the 
modified  shape  for  near  objects.  It  will  be  noticed  that  the 
alteration  is  almost  confined  to  the  anterior  portion  of  the  lens. 
The  capacity  for  adjusting  the  power  of  the  lens  to  the 
immediate  demands  upon  it  is  very  remarkable  in  young 
children,  enabling  them  to  see  objects  with  perfect  sharpness 
from  a  distance  of  three  or  four  inches  up  to  an  indefinitely 
great  distance;  but  throughout  life  it  suffers  a  diminution 


158 


LIGHT 


which  seems  to  follow  a  fairly  constant  rate.  If  during 
the  earlier  period  of  life  the  eye  is  used  almost  exclusively 
for  near  objects,  the  lens  is  apt  to  assume  a  permanently 
thickened  form,  so  that  it  is  too  powerful  for  waves  having  a 
remote  source.  Such  an  eye  is  called  near-sighted,  or  myopic, 


FIGURE  41. 

and  in  order  to  see  distant  objects  well  its  power  must  be 
reduced  by  means  of  a  negative  lens,  that  is,  of  a  spectacle 
glass  thinner  in  the  middle  than  at  the  edge.  This  is  the 
common  origin  of  the  near-sightedness  which  frequently 
makes  its  appearance  during  youth  among  those  of  studious 
or  sedentary  habits.  If  this  predisposing  cause  of  myopia 
is  wanting,  the  gradual  loss  of  the  power  of  accommodation 
with  advancing  age  results  in  the  adjustment  becoming  more 
and  more  close  than  proper  for  remote  objects.  An  eye 
with  this  sort  of  limitation  is  called  far-sighted,  or  presbyopic. 
To  adapt  it  to  clear  vision  of  near  objects  its  power  must  be 
increased  by  the  aid  of  a  positive  lens,  that  is,  one  which  is 
thicker  in  the  middle  than  at  the  edge.  The  change  will 
generally  have  progressed  so  far  by  the  fifth  decade  of  one's 
life  that  the  nearest  point  of  distinct  vision  is  at  arm's 
length,  when  the  necessity  of  aids  to  vision  becomes  first 
evident  even  to  those  who  are  very  unobservant.  One  or  the 
other  of  the  foregoing  descriptions  is  the  common  history  of 
the  normal  eye,  but  many  persons  are  born  with  myopic  eyes, 
generally  on  account  of  abnormal  axial  length  of  the  eye- 


THE  EYE  159 

ball ;  likewise,  instances  are  not  rare  where  a  defect  of  the 
opposite  kind,  too  short  an  axis,  is  congenital.  In  the  latter 
case  positive  lenses  will  be  required  for  seeing  even  remote 
objects  clearly.  We  may  note,  in  passing,  that  another  de- 
fect, unfortunately  far  from  uncommon,  is  that  in  which  the 
eye  has  different  powers  in  different  planes.  This  can  be 
recognized  by  the  differing  distinctness  with  which  horizontal 
and  vertical  lines  may  be  seen  in  a  brick  wall,  or  by  the 
elongated  aspect  of  a  bright  star.  The  defect  is  known  as 
astigmatism,  and  it  may  be  corrected  by  the  use  of  a  spec- 
tacle glass  bounded  by  cylindrical  surfaces. 

The  rear  portion  of  the  inner  eye  is  immediately  sur- 
rounded by  a  thin,  black,  membranous  lining  between  the 
retina  and  the  firm  coat  (sclerotic  coat)  which  forms  the  body 
of  the  eye,  the  office  of  which  is  to  shield  the  retina 
from  all  light  not  entering  through  the  pupil.  This  dark 
membrane  is  named  the  choroid.  It  is  evident  that  the  pupil 
appears  quite  black  to  an  observer  because  of  this  membrane 
as  well  as  because  of  the  fact  that  the  only  points  of  the 
retina  visible  to  such  an  observer  are  those  covered  by  the 
image  of  his  own  pupil.  The  first  condition,  namely, 
the  presence  of  the  choroid,  is  wanting  in  albinos,  and  the 
second,  a  sharp  retinal  image,  in  animals  whose  vision  is 
imperfect  or  who  have  their  eyes  adjusted  for  something 
widely  remote  from  the  spectator.  In  these  cases  the  pupil 
no  longer  seems  black  but  of  the  pink  color  of  the  retina 
behind,  and  a  singular  glare  appears  which  is  sometimes 
thought  to  have  its  origin  in  the  eye  itself.  It  is  easy  to 
devise  means  of  illuminating  the  retina  by  placing  either  a 
transparent  mirror  between  the  eye  of  the  observer  and  the 
observed  eye,  or  a  mirror  with  a  small  perforation  which 
permits  the  observer  to  look  through  it.  Instruments  for 
this  end  are  called  ophthalmoscopes,  and  are  of  great  impor- 
tance to  the  surgeon. 

The  blood-vessels  which  serve  to  nourish  the  retina  lie  on 
its  anterior  surface,  while  the  portions  that  are  the  true 
sensory  organs  are  confined  to  the  posterior  surface.  For  this 


160  LIGHT 

reason  light  which  enters  the  eye  from  any  point  produces 
shadows  of  these  vessels  in  the  retina.  These  are  usually 
unrecognized,  both  from  their  ordinarily  ill-defined  nature  on 
account  of  the  angular  dimensions  of  the  pupil,  and  because 
the  portions  of  the  retina  immediately  behind  these  vessels, 
consequently  those  most  frequently  affected,  have  adjusted 
themselves  to  the  relative  darkening  due  to  their  presence. 
If,  however,  one  looks  at  the  bright  sky  through  a  pin-hole 
in  a  card,  or  if  an  image  of  a  bright  object  at  the  extreme 
margin  of  the  field  of  vision  sends  its  light  down  upon  the 
retina,  the  shadows  become  surprisingly  obvious,  especially 
if  the  source  of  light  is  rendered  intermittent,  or,  in  the 
former  experiment,  if  the  card  is  kept  in  constant  motion. 
An  admirable  way  to  secure  the  effect  is  by  means  of  a  lens 
to  form  an  image  of  a  brilliant  light  on  the  ball  of  the 
eye,  just  back  of  the  edge  of  the  cornea;  the  light  which 
penetrates  the  sclerotic  coat  will  produce  wonderfully  distinct 
shadows. 

Just  opposite  the  centre  of  the  pupil  at  the  back  of  the  eye 
there  is  a  slightly  depressed  portion  (5,  Figure  40)  called  the 
yellow  spot,  or  macula  fovea.  This  is  the  region  of  most 
accurate  vision,  and  visual  impressions  derived  from  images 
not  restricted  to  this  spot  are  always  somewhat  ill  defined. 
Our  knowledge  of  the  color  has  been  derived  primarily  from 
dissection  of  the  eye,  but  an  interesting  evidence  of  its 
nature  may  be  found  by  putting  a  piece  of  blue  glass  over 
the  pin-hole  in  the  perforated-card  experiment,  when  it  will 
be  found  that  this  spot,  which  lies  at  the  precise  centre  of 
the  field  of  vision,  is  indicated  not  only  by  the  relative  free- 
dom from  blood-vessels,  but  also  by  its  distinctly  darker 
shade.  The  latter  distinction  is  due  to  the  greater  absorption 
of  blue  light  by  this  portion  of  the  retina,  thus  demonstrating 
its  yellow  color. 

The  region  at  which  the  optic  nerve  enters  the  back  of 
the  eye  is  called  the  white  spot,  or  macula  lutea.  This  spot, 
occupying  a  very  considerable  area  in  the  visual  field,  is 
wholly  insensitive  to  light,  and  consequently  marks  a  blind 


THE  EYE  AND   VISION  161 

spot  in  the  field.  That  every  human  eye  possesses  an  area 
which  is  quite  blind  and  is  large  enough  to  render  invisible 
an  area  seventeen  times  as  large  as  that  covered  by  the  full 
moon,  or  as  large  as  would  be  occupied  by  the  image  of 
the  head  of  a  man  at  a  distance  of  six  or  seven  feet,  would 
doubtless  surprise  most  people  not  informed  in  this  depart- 
ment of  science.  In  the  year  1668  Mariotte,  while  en- 
deavoring to  find  whether  the  optic  nerve  proper  is  sensitive 
to  light  —  a  scientific  question  which  seems  to  have  occurred 
to  no  one  before  him  —  made  this  unexpected  discovery, 
which  excited  a  very  lively  popular  interest  at  the  time.  It 
is  easy  to  demonstrate  the  existence  and  position  of  this  spot 
by  means  of  a  card  having  a  mark  and  an  appropriate  object, 


FIGURE  42. 

as  illustrated  in  Figure  42.  If  in  this  the  right  eye  is  fixed 
upon  the  cross,  the  left  eye  being  closed,  and  the  distance 
of  the  figure  from  the  eye  is  varied,  it  is  easy  to  find  a  posi- 
tion at  which  the  large  white  circle  entirely  disappears, 
although  one  can  still  see  the  regions  surrounding  it.  A 
few  persons  can  recognize  the  existence  and  position  of  the 
blind  spot,  without  the  aid  of  a  definite  figure,  by  careful 
attention  to  the  visual  field  when  one  eye  is  closed. 

Experiment  shows  that  the  seat  of  visual  sensation  is  re- 
stricted to  an  extremely  thin  layer  at  the  back  of  the  retina ; 
therefore  the  ultimate  structure  of  the  nerve  tissue  in  this 
region  is  of  especial  interest.  A  powerful  microscope  shows 
that  it  is  composed  of  two  types  of  elements,  which  are  repre- 

11 


162 


LIGHT 


FlGUKE   43. 


sented  in  Figure  43,  one  of  which  (a)  is  called  a  rod  and 
the  other  (6)  is  called  a  cone.  We  are  to  understand  that 

the  whole  sensitive  portion  of  the 
retina  consists  of  rods  and  cones 
radially  disposed  with  respect  to 
the  centre  of  the  eye;  but  it  is 
found  that  within  the  yellow  spot 
only  cones  are  present,  while  these 
elements  become  relatively  few  as 
one  approaches  the  peripheral  por- 
tions of  the  retina.  For  this  reason, 
together  with  the  fact  that  the  eye 
is  found  to  be  more  sensitive  to 
feeble  illumination  in  regions  out- 
side the  yellow  spot,  it  is  assumed 
that  the  cones  have  to  do  with  the 
minute  perception  of  form,  and  the 
rods  with  perception  of  differences 
of  illumination.  This  inference  is 

strengthened  by  the  observed  fact  that  the  angular  distances 
of  adjacent  cones  correspond  fairly  accurately  with  the  mini- 
mum angular  distance  at  which  two  objects  can  be  seen  other 
than  single.  This  distance,  one  of  the  most  important  con- 
stants of  the  eye,  is  about  one  minute  of  arc,  as  has  been 
already  pointed  out  on  page  95. 

Many  visual  phenomena  depend  on  the  fact  that  the  sensi- 
tiveness of  the  retina  falls  off  with  remarkable  rapidity  after 
the  beginning  of  excitation.  For  example,  after  remaining  in 
darkness  for  a  long  time  the  light  of  the  full  moon  produces 
a  painfully  strong  impression,  but  under  other  circumstances 
the  light  of  a  bright  noonday  may  be  less  impressive  although 
620,000  times  as  intense.  A  retinal  impression  lasts  for  a 
perceptible  time  after  the  cessation  of  the  stimulus,  however, 
as  appears  from  the  fact  that  a  rapidly  moving  bright  point 
seems  to  carry  with  it  a  streak  of  light ;  and  if  the  path  is  a 
closed  curve  in  which  the  point  travels  as  frequently  as 
ten  or  more  times  a  second,  the  streak  seems  to  be  also  a 


THE  EYE  AND   VISION  163 

closed  curve  of  nearly  uniform  brightness.  Such  experi- 
ments prove  that  the  sensation  endures  without  very  great 
diminution  for  a  time  as  long  as  a  tenth  of  a  second.  Images 
which  appear  upon  the  retina  after  the  cessation  of  the 
light  producing  them  are  called  after-images.  Very  intense 
after-images  often  last  many  seconds,  but  they  undergo  a 
succession  of  changes  in  color  and  intensity  that  are  highly 
complex,  and  which  cannot  be  regarded  as  belonging  to 
strictly  normal  vision.  Such  images  are  distinguished  from 
the  former  kind  by  the  term  negative  after-images,  and 
although  their  study  has  proved  of  value  in  several  fields  of 
physiological  optics,  they  will  not  be  further  considered  here. 

There  is,  strictly  speaking,  no  such  thing  as  an  optical 
centre  to  the  eye,  but  there  are  two  points  separated  by  an 
interval  of  about  a  sixtieth  of  an  inch,  so  related  that  light 
which  would  pass  through  the  first  before  refraction  appears 
to  pass  through  the  second  without  change  of  direction  after 
refraction.  These  points  are  called  the  first  and  second  nodal 
points,  respectively.  A  point  half-way  between  them  may  be 
taken  in  most  instances  as  the  optical  centre  of  the  eye,  so 
that  light  having  a  direction  which  would  carry  it  through 
this  point  will  remain  unchanged  in  direction  after  its  final 
refraction.  Since  this  centre  is  well  in  front  of  the  geomet- 
rical centre  of  the  eye  about  which  it  rotates,  a  distant  object 
just  hidden  behind  a  screen  close  to  the  eye  becomes  visible 
by  oblique  vision  when  the  eye  is  turned  away  from  its  direc- 
tion, even  when  the  head  remains  fixed. 

As  there  is  no  provision  for  eliminating  the  secondary 
phenomenon  of  dispersion  which  accompanies  all  cases  of 
refraction,  the  eye  is  not  achromatic.  Although  ordinarily 
overlooked,  it  is  not  at  all  difficult  to  devise  experiments 
which  render  this  defect  conspicuous.  For  example,  should 
half  the  pupil  be  covered  by  an  opaque  screen,  the  image  of 
a  linear  object  seen  against  a  bright  field — a  window  bar 
against  the  sky,  for  example  —  if  parallel  to  the  edge  of 
the  screen,  will  appear  blue  on  one  side  arid  yellow  on  the 
other.  But  the  obvious  effects  due  to  this  imperfection  in 


164  LIGHT 

the  eye  are  noted  when  observing  a  surface  having  figures 
of  bright  red  color  upon  a  bright  green  or  blue  field ;  in  this 
case  a  peculiarly  disagreeable  sensation  arises  from  unsuc- 
cessful efforts  to  accommodate  for  both  colors  at  the  same 
time. 

A  structural  defect  in  the  eyes  gives  rise  to  certain  visual 
phenomena  even  more  conspicuous  than  the  chromatic  errors 
just  noted,  that  is,  a  lack  of  homogeneity  in  the  lens  itself. 
This  body,  built  up  of  transparent  cells  in  a  highly  complex 
manner,  has  a  roughly  stellate  structure,  which,  even  in  the 
best  eyes,  is  the  immediate  cause  of  the  irregularly  pointed 
aspect  of  bright  stars.  No  one  of  these  defects  seriously 
impairs  vision,  provided  that  the  accommodation  is  perfect; 
but  when  this,  too,  is  at  fault,  the  light  from  a  point,  instead 
of  being  uniformly  distributed  over  a  small  portion  of  the 
retina,  is  somewhat  irregularly  gathered  in  areas  radially 
placed  in  accordance  with  the  lens  structure.  It  is  for  this 
reason  that  the  horns  of  a  crescent  moon  appear  multiple 
unless  the  eye  is  perfectly  adjusted  as  regards  accommoda- 
tion. Other  defects,  which  vary  largely  with  different  indi- 
viduals, are  found  in  the  presence  of  opaque  bodies  within 
the  media  of  the  eye,  which,  although  they  cast  shadows 
upon  the  retina,  are  ordinarily  overlooked  on  account  of 
the  quickly  acquired  insensitiveness  under  continuous  im- 
pression; still,  they  can  always  be  detected  if  the  eye  is 
turned  suddenly  from  the  contemplation  of  a  dark  field  to 
a  bright  sky.  These  evanescent  objects  are  called  muscae 
volitantes. 

If  in  a  darkened  room  the  eyes  are  closed  and  the  eyeball 
is  slightly  compressed  with  the  finger-tip  at  a  point  as  far 
removed  from  the  pupil  as  possible,  a  deep  violet-colored 
spot  surrounded  by  a  ring  of  bright  light  is  recognized  in 
that  portion  of  the  visual  field  which  corresponds  to  the  part 
of  the  retina  thus  disturbed.  Again,  if  in  a  dark  room  the 
eyes  are  turned  suddenly  as  far  as  possible,  either  to  the 
right  or  left,  similar  spots  will  be  seen  which  are  produced 
by  a  strain  brought  upon  the  retina  by  the  stretching  of  the 


COLOR  SENSATION  165 

optic  nerves.  These  observations  are  interesting  illustra- 
tions of  the  important  physiological  law  that  an  excitation  of 
a  sense  organ  results  in  some  kind  of  sensation  proper  to 
that  organ,  and  is  quite  independent  of  the  nature  of  the 
stimulus.  In  the  cases  described,  of  course,  there  can  be  no 
possible  analogy  between  the  ordinary  stimulus  produced  by 
light  and  this  mechanical  disturbance ;  so,  too,  singular  light 
sensations,  and  light  sensations  only,  result  when  an  electric 
current  is  directed  through  the  eye.  The  subjective  phe- 
nomenon known  as  "seeing  stars  "  is  produced  by  a  disturb- 
ance of  the  elements  of  the  retina  attending  a  violent  shock, 
although  it  may  be  perhaps  due  to  a  sort  of  reflex  action 
from  the  brain. 

The  study  of  color  sensations  offers  a  field  of  great  inter- 
est, in  which  Sir  Isaac  Newton  made  the  earliest  arid  most 
important  investigations.  He  demonstrated  that  light  of 
different  refrangibilities  produces  different  color  sensations, 
and  that  white  light  can  be  decomposed  by  a  prism  so  as  to 
give  rise  to  an  infinite  number  of  color  sensations.  To  seven 
of  these  he  attached  familiar  color  names.  He  also  showed 
that  these  seven  colors  combined  in  proper  proportions  would 
cause  the  sensation  of  white  light.  He  even  went  further 
than  this,  and  proved  that  if  the  spectral  series  of  colors 
is  expanded  by  the  addition  of  purple,  it  is  always  possible, 
having  chosen  one  color,  to  find  a  second  which,  combined 
with  it,  would  produce  white  light.  Pairs  of  colors  so  related 
are  called  complementary  colors.  Although  Newton's  con- 
tribution to  the  theory  of  color  sensation  is  by  no  means 
restricted  to  the  discoveries  named,  we  shall  find  it  more  con- 
venient to  consider  the  general  theory  developed  in  modern 
times. 

Critical  study  of  color  sensations  has  led  modern  writers 
on  chromatics  to  revise  Newton's  terminology  of  the  spectral 
colors  in  replacing  his  seven  by  the  series  red,  orange,  yel- 
low, green,  blue-green,  blue,  violet.  This  rejects  Newton's 
indigo  and  supplies  blue-green,  or  robin's  egg  color,  which 
is  not  only  quite  as  properly  an  independent  color  as  is 


166  LIGHT 

orange,  but  is  very  convenient  because  exactly  complemen- 
tary to  vermilion  red.  Since  Newton  had  proved  that  with 
two  complementary  colors  it  is  possible  to  secure  by  combina- 
tion, not  only  either  one  of  the  colors,  but  also  any  gradation 
between  it  and  white,  and  that  with  three  colors  it  is  there- 
fore possible  to  produce  a  far  greater  range  of  colors,  the 
question  naturally  arises  as  to  the  minimum  number  of  pri- 
mary colors  by  the  use  of  which  all  possible  colors  might  be 
imitated;  or,  to  state  the  problem  in  a  much  more  scientific 
form  —  as  color  is,  after  all,  a  question  of  sensation  only  — 
what  is  the  smallest  number  of  primary  color  sensations  that 
by  appropriate  combination  will  yield  all  possible  color  sensa- 
tions? Dr.  Thomas  Young  seems  first  to  have  stated  this 
problem  clearly  and  to  have  given  its  solution.  His  assump- 
tions are  the  following :  — 

I.  The  eye  possesses  three  kinds  of  nerve  termini.     Ex- 
citation of  the  first  gives  rise  to  the  sensation  of  red,  of  the 
second  to  the  sensation  of  green,  and  of  the  third  to  the 
sensation  of  violet. 

II.  Monochromatic  light,  that  is,  light  of  a  single  wave- 
length,   stimulates   these   three   kinds   of  nerve   termini   in 
differing  ratios,  with  different  lengths  of  the  waves.     The 
nerves  sensitive  to  red  will  be  most  strongly  affected  by  light 
of  greater  wavelength,  the  green-sensitive  nerves  by  light  of 
medium  length  of  waves,  and  the  last  by  the  shortest  waves. 
It  is  not  assumed,  however,  that  any  spectral  color  fails  to 
excite  one  or  more  of  the  nerve  elements ;  on  the  contrary,  in 
order  to  explain  a  vast  number  of  phenomena,  it  is  necessary 
to  assume  that  every  spectral  color  excites  simultaneously  all 
the  elements,  but  to  a  varying  degree. 

To  render  clear  the  second  of  the  fundamental  assump- 
tions we  may  refer  to  Figure  44,  which  is  copied  from  Helm- 
hoi  tz.  In  it  the  letters  at  the  bottom  represent  the  spectral 
colors,  and  the  curves  above  them  the  relative  sensitiveness 
of  the  three  hypothetical  nerve  elements ;  whence  we  deduce 
that  simple  red  stimulates  the  red-sensitive  nerves  strongly 
and  the  others  feebly,  with  a  resulting  sensation  of  red,  and 


COLOR   SENSATION 


167 


simple  yellow  excites  moderately  strongly  the  red  and  green 
elements  and  feebly  the  violet,  with  a  sensation  of  yellow  as 
a  result.  Other  colors  may  be  discussed  in  a  similar  way. 
It  is  proper  to  remark  that  the  figure  carries  the  term  blue 
instead  of  violet  in  the  lowest  curve,  but  this  is  not  material 
in  the  theory,  nor  is  it  yet  determined  which  is  preferable. 
We  shall  hereafter  employ  the  convenient  although  unscien- 
tific terms,  red  nerves,  green  nerves,  etc.,  to  designate  these 
elements,  which  are  at  the  foundation  of  the  theory. 


This  is  a  sketch  of  the  famous  theory  known  to  physicists 
and  physiologists  as  the  Young-Helmholtz  Theory  of  Color 
Sensation.  On  account  of  its  celebrity,  its  simplicity,  and 
its  general  agreement  with  experience,  we  shall  describe  it 
somewhat  at  length,  although  it  is  quite  certain  that  it  must 
be  regarded  only  as  a  first  approximation  to  the  explanation 
of  a  highly  complex  series  of  phenomena. 

An  obvious  corollary  from  the  theory  is  that  the  normal 
eye  never  has  a  given  fundamental  color  sensation  without  at 
the  same  time  having  in  a  less  degree  a  combination  of  the 
other  two.  For  example,  vivid  green  light  falling  upon  the 


168  LIGHT 

retina  awakens  a  moderate  sensation  of  a  mixture  of  red  and 
blue  with  the  stronger  green,  which  we  may  regard  as  com- 
bining with  a  portion  of  the  green  sensation  to  produce 
white ;  hence  the  sum  of  the  sensations  may  be  characterized 
as  a  pure  green  mixed  with  a  certain  amount  of  white.  It 
follows  from  this  that  were  it  possible  to  stimulate  the 
green  nerves  alone  the  resulting  sensation  of  greenness  would 
be  far  more  intense.  In  effect  this  result  can  be  readily 
produced  by  taking  advantage  of  the  rapid  loss  of  sensitive- 
ness of  the  retina  under  stimulus.  If  one  looks  for  a  few 
minutes  at  a  sheet  of  magenta  paper,  which  is  a  nearly  equal 
combination  of  red  and  of  blue,  the  eye  will  have  lost  a  con- 
siderable part  of  its  sensitiveness  to  these  two  colors,  so  that 
if  green  light  falls  upon  the  modified  retina,  the  sensation  of 
greenness  will  be  astonishingly  enhanced.  The  same  prin- 
ciple of  course  holds  true  for  all  pairs  of  complementary 
colors,  that  is,  either  one  of  such  a  pair  would  appear  with 
increased  strength  of  color  after  the  retina  has  been  slightly 
wearied  by  contemplating  the  other. 

Before  applying  the  foregoing  theory  to  the  explanation  of 
colors  we  must  acquaint  ourselves  with  certain  useful  terms 
employed  in  the  discussion  of  chromatics,  and  with  some  of 
the  experimental  methods  of  testing  the  conclusions  deduced 
from  the  theory.  It  has  been  found  that  all  color  sensations 
can  be  defined  by  the  use  of  three  terms  only :  The  first  is 
hue,  that  is  to  say,  that  which  gives  the  ordinary  color  name ; 
the  second  is  saturation,  or  freedom  from  admixture  of  white 
light;  the  third  is  luminosity,  which  is  the  equivalent  of 
brightness  when  applied  to  white  light.  A  few  illustrations 
derived  from  common  color  names  will  render  clear  the  mean- 
ings of  these  important  terms.  A  straw-color,  for  example, 
has  the  hue  yellow,  the  saturation  very  low,  and  the  lumi- 
nosity very  high;  olive-green  has  a  greenish  yellow  hue  of 
high  saturation  and  low  luminosity;  pink  is  a  rose-red  hue 
of  low  saturation  and  high  luminosity.  These  examples  are 
perhaps  enough  to  serve  our  present  purpose,  but  many 
others  will  appear  later. 


COLOR  SENSATION  169 

We  may  now  turn  to  a  consideration  of  some  of  the  methods 
employed  in  experimenting  with  these  mixed  color  sensations, 
for  it  will  be  observed  that  the  whole  theory  rests  upon  the 
assumption  that  the  retina  is  stimulated  by  the  simultaneous 
action  of  unlike  colored  lights  —  a  very  different  thing 
from  the  action  of  light  coming  from  a  mixture  of  unlike 
pigments.  Thus  we  shall  see  presently  that  the  combination 
of  yellow  and  blue  lights  may  produce  a  sensation  of  pure 
whiteness  —  would  be,  in  short,  white  light  —  but  it  is  a  fact 
known  to  every  child  that  light  from  a  mixture  of  yellow  and 
blue  paints  is  in  general  green.  The  reason  for  this  singular 
difference,  which  so  often  leads  to  a  quite  false  conception  of 
the  requirements  of  a  sound  color  theory,  will  appear  when 
we  come  to  consider  the  cause  of  color  in  natural  objects. 
Any  experimental  tests  of  the  theory,  therefore,  must  be 
made  by  an  actual  mixture  of  the  colored  lights  in  question. 
This  may  be  evidently  done  by  reflecting  light  from  one 
source  on  to  a  screen  which  is  illuminated  by  light  from  a 
second  source  at  the  same  time,  and  almost  as  simply  in  a 
number  of  other  ways ;  but  none  is  so  convenient  and  of  such 
universal  applicability  as  the  method  invented  by  Maxwell. 
The  principle  at  the  basis  of  this  method  is  the  persistence 
of  visual  impressions.  Imagine  a  disk  of  cardboard  half  of 
which  is  covered  with  yellow  paper  and  the  other  half  with 
blue  paper;  if  this  is  rotated  about  its  centre  many  times 
a  second,  it  would  appear  to  the  eye  uniformly  tinted  with 
yellow  of  half  the  luminosity  possessed  by  the  yellow  paper 
added  to  blue  of  half  the  luminosity  which  would  be  pre- 
sented by  a  disk  entirely  covered  with  the  blue  paper,  and 
the  combination  would  be  found  free  from  all  color,  that  is, 
it  would  be  white.  But  this  white  would  clearly  be  of  a 
low  luminosity;  in  short,  in  comparison  with  white  paper,  it 
would  be  called  gray.  If  we  have  two  cardboard  disks,  each 
possessing  a  radial  slit  and  a  uniform  color,  the  two  may  be 
put  together  so  that  either  one  will  exhibit  a  sector  of  the 
circle  as  large  as  desired.  This  composite  disk  rotated 
rapidly  about  its  centre  will  enable  us  to  get  the  result  of 


170  LIGHT 

the  combination  of  the  two  colors  in  any  required  and  easily 
determined  ratio.  The  defect  of  the  method  lies  in  the  fact 
that  all  such  compound  colors  possess  a  luminosity  less  than 
the  brightest  of  the  components,  but  the  convenience  is  so 
great  that  it  offers  the  best-known  method  of  exhibiting  the 
principal  phenomena.  Small  disks  of  colored  papers  not 
more  than  an  inch  in  diameter  dropped  axially  upon  a  spin- 
ning top  are  admirably  adapted  for  the  experiments  to  be 
described. 

The  choice  of  the  three  fundamental  colors  is  to  a  certain 
extent  arbitrary  —  the  only  condition  imposed  is  that  it  shall 
be  possible  to  produce  white  by  a  proper  combination  involv- 
ing all  three  —  but  Maxwell's  choice  of  vermilion-red,  emerald- 
green,  and  ultramarine-blue,  all  of  which  may  be  readily  pro- 
cured, has  much  to  recommend  it;  and  various  phenomena 
presented  by  color-blind  eyes  render  it  practically  certain 
that  these  are  not  far  out  of  the  way.  If,  according  to  the 
method  described,  red  and  blue  are  combined  in  varying 
proportions,  starting  with  the  whole  of  the  red  disk  show- 
ing and  ending  with  this  completely  covered  with  the  blue 
one,  it  will  be  found  that  we  have  a  series  of  hues  pass- 
ing continuously  from  pure  red  through  rose-red,  magenta, 
and  violet,  to  pure  blue;  if  the  red  disk  is  replaced  by  a 
green  one  and  a  similar  experiment  is  made,  we  pass  from 
blue  through  the  hues  greenish  blue,  blue -green,  bluish  green, 
to  pure  green;  finally,  if  the  blue  disk  is  replaced  by  the 
red  one  and  a  similar  progressive  change  is  made,  we  pass 
through  a  series  of  colors  in  the  order  yellowish  green,  green- 
ish yellow,  yellow,  orange,  to  a  pure  red.  Should  all  three 
disks  be  employed  at  once,  it  will  be  found  quite  easy  to 
arrange  them  so  that  the  combined  impression  is  a  pure  gray ; 
in  other  words,  a  white  of  low  luminosity. 

This  series  of  experiments  teaches  us  that  it  is  possible  to 
secure  all  hues  by  a  properly  chosen  combination  of  the  three 
fundamental  colors,  but  it  is  very  far  from  presenting  us 
examples  closely  approximating  many  of  the  most  familiar 
colors.  We  have,  it  is  true,  secured  all  possible  hues,  but 


COLOR  DIAGRAM 


171 


only  a  small  part  of  all  possible  colors.  The  distinction  lies 
in  the  fact  that  we  control  by  this  means  only  the  relative 
proportions  of  colored  lights,  whereas  the  other  two  elements 
defining  a  color  sensation  are  not  varied  at  will ;  hence  this 
general  review  of  colors  is  very  restricted.  It  may,  however, 


FIGURE  45. 

be  greatly  extended  by  the  use  of  a  diagram  invented  for  this 
purpose.  Suppose  we  place  the  three  fundamental  colors  at 
the  vertices  of  an  equilateral  triangle,  as  represented  in 
Figure  45,  in  which  these  colors  are  designated  by  their 
initial  letters,  and  select  the  centre  as  the  point  to  indicate 
white;  then,  if  we  make  the  assumption  that  any  point  on 


172  LIGHT 

one  of  the  sides  of  the  triangle  represents  a  hue  produced  by 
mixing  the  two  adjacent  primary  colors  in  an  inverse  ratio  to 
the  distance  of  the  point  from  either  of  the  corresponding 
apexes,  we  have  an  absolutely  unambiguous  method  of  de- 
fining all  possible  hues.  If  in  addition  to  this  we  let, 
conventionally,  the  linear  distance  from  the  central  point 
represent  the  saturation,  or  freedom  from  admixture  of  white, 
every  point  in  the  plane  of  the  triangle  will  represent  a  color 
of  which  two  of  the  three  necessary  constants  are  given  in  a 
perfectly  definite  manner.  A  number  attached  to  a  given 
point  in  this  diagram  may  be  used  to  designate  the  lumi- 
nosity. From  such  a  diagram  much  may  be  learned.  For 
example,  it  gives  at  once  all  pairs  of  complementary  colors, 
since  any  straight  line  drawn  through  the  central  point  cuts 
the  sides  of  the  triangle  in  two  points  which  define  colors 
thus  related.  Suppose  one  demands  the  color  complementary 
to  emerald-green;  the  diagram  shows  it  to  be  a  color  on  a 
line  passing  through  G-  and  W  and  on  the  opposite  side  of 
the  latter  point;  all  this  region  represents  a  hue  which 
is  an  equal  mixture  of  red  and  blue,  called  magenta  when 
highly  saturated  and  lilac  when  pale.  The  region  from 
green  to  blue-green  would  find  its  complementary  hues  be- 
tween magenta  and  pure  red,  while  those  on  the  other  side 
of  the  green  would  be  complementary  to  hues  lying  between 
magenta  and  ultra  marine -blue,  that  is  to  say,  to  the  purples 
and  violets.  A  line  drawn  from  B  through  W  intersects  the 
R&  line  near  its  middle  point,  the  region  of  yellow.  It  is 
just  this  portion  of  the  diagram  which  appears  to  be  least 
satisfactory  as  a  means  of  representing  familiar  colors,  and 
it  is  therefore  worthy  of  more  careful  attention.  According 
to  the  theory  an  equal  mixture  of  red  and  green  should  form 
a  yellow,  but  an  experiment  after  the  Maxwell's  disks  method 
gives  a  dingy  brownish  yellow  color  of  which  the  relation  to 
the  familiar  bright  yellow  so  extremely  common  in  flowers 
is  by  no  means  obvious ;  and  it  is  quite  impossible  to  secure 
a  good  yellow  by  this  means.  This  is  because  the  luminosity 
of  yellow  is  very  great,  whereas  light  from  green  paper  is 


COLOR   COMBINATIONS  173 

much  less  bright  and  that  from  red  paper  is  far  more  inferior. 
There  is  no  difficulty  in  demonstrating  the  identity  of  hue  of 
the  mixed  color  and  the  bright  yellow,  however,  by  a  some- 
what indirect  method.  If  we  take  three  smaller  disks,  one 
white,  one  black,  and  the  third  yellow,  and  superimpose  them 
upon  the  larger  disks  of  red  and  green,  it  is  always  possible 
to  adjust  the  former  so  that  in  rapid  rotation  the  inner  disk 
matches  the  outer  circle.  Figure  46  illustrates  this  interest- 
ing experiment,  which  proves 
that  the  yellow  produced  by 
the  combination  of  the  funda- 
mental red  and  green  is  less 
saturated  than  that  of  the 
bright  yellow,  since  it  was 
found  necessary  to  add  white 
to  the  latter;  again,  the  mixed 
color  was  much  less  luminous 
because  it  was  found  necessary 
to  reduce  the  brightness  of  the 

.  FIGURE  46. 

yellow  paper  by  the  inclusion  of 

a  black  sector  in  order  to  secure  a  satisfactory  match.  We 
may  therefore  make  the  deduction  that,  in  the  diagram  of 
Figure  45,  the  place  for  the  color  which  bears  the  name, 
unqualified,  of  yellow  is  on  the  line  drawn  from  W  through 
the  mid-point  of  RG-,  but  two  or  three  times  as  distant  from 
the  latter  point.  The  extraordinary  luminosity  of  yellow, 
unequalled  by  any  other  color  of  as  high  a  degree  of  satura- 
tion, is  explained  by  a  prismatic  analysis  of  the  color,  which 
shows  that  such  yellow  is  merely  white  deprived  of  its  blue 
and  violet  components,  both  of  which  are  of  relatively  feeble 
luminosity.  This  experiment  also  proves  the  converse,  that 
the  colors  last  named  are  relatively  strong  colors,  that  is, 
that  a  small  admixture  of  either  of  them  would  produce  an 
inordinately  great  change  in  the  hue.  In  accordance  with  the 
latter  deduction  the  spectrum  of  a  deep  blue  sky  exhibits 
astonishingly  little  difference  from  that  of  a  white  cloud. 
Although  a  large  number  of  names  has  appeared  in  the 


174  LIGHT 

discussion  of  the  color  diagram,  there  are  certain  groups 
of  names  which  have  not  appeared  at  all;  such  are  the 
browns,  olives,  lavenders,  etc.  For  a  physical  definition  of 
these  terms  we  must  look  to  variation  in  the  elements  of 
luminosity  and  of  saturation.  A  simple  experiment  will 
show  that  any  hue  lying  between  red  and  yellow,  of  a  high 
degree  of  saturation  and  a  very  low  degree  of  luminosity,  is 
a  brown.  Hues  from  greenish  yellow  to  yellowish  green 
with  the  same  characteristics  yield  the  group  of  olives. 
These  statements  may  be  readily  tested  by  putting  a  small 
sector  of  bright-colored  paper  of  the  hue  in  question  upon  a 
disk  of  black  paper  and  rotating  it  rapidly.  By  a  similar 
experiment  with  any  hue  from  pure  green  to  magenta  and 
even  somewhat  beyond,  we  meet  with  no  new  color  terms, 
but  content  ourselves  with  the  prefix  "dark"  as  sufficiently 
descriptive.  Now  vary  the  experiment  by  placing  the  nar- 
row sectors  of  colored  papers  over  white  disks  and  set  them 
in  rotation;  we  shall  then  find  colors  which  would  be  char- 
acterized as  pale  red,  pale  yellow,  pale  green,  etc.,  until, 
having  passed  the  region  of  pale  blue,  we  find  that  ultra- 
marine-blue thus  diluted  yields  a  lavender,  while  further 
progress  in  the  same  direction  gives  lilacs  and  pinks.  It  is 
an  odd  fact  that  in  the  first  region  described  new  color  terms 
appear  only  when  the  luminosity  is  diminished  and  not 
when  the  saturation  is  decreased,  while  in  the  second  region 
the  exact  contrary  holds  true.  There  is  undoubtedly  a  rea- 
son for  this,  founded  upon  the  psychology  of  color  sensations, 
but  it  would  probably  be  difficult  to  formulate  it. 

The  natural  colors  of  objects  are  due  to  their  property  of 
reflecting  only  a  portion  of  the  light  waves  falling  upon 
them,  and  a  portion  differing  greatly  with  different  wave- 
lengths. There  are  a  few  substances  in  which  this  action  is 
confined  to  the  surface,  such  as  colored  metals  and  a  variety 
of  less  well-known  things  which  we  describe  as  possessing 
a  metallic  color.  Certain  aniline  dyes  in  the  solid  state  are 
good  examples  of  the  latter  group  of  bodies.  But  in  a  vast 
majority  of  cases  the  modification  of  the  reflected  light  is 


NATURAL   COLORS  OF  OBJECTS  175 

produced  by  absorption  within  the  material  of  the  body. 
The  light  reflected  from  the  surface  of  a  ruby  or  of  a  sapphire 
is  as  white  as  that  reflected  from  a  diamond,  but  such  light 
as  has  passed  through  any  portion  of  the  former  gems  appears 
of  the  characteristic  color  on  account  of  the  suppression  of 
certain  portions  of  the  white  light  by  absorption.  Most 
pigments  are  simply  small  fragments  of  highly  colored  crys- 
tals imbedded  in  a  transparent  medium.  Chrome  yellow,  for 
example,  when  used  as  a  paint,  consists  of  finely  powdered 
crystals  of  lead  chromate  mixed  with  oil.  If,  by  means  of 
a  prism,  we  examine  light  which  has  been  modified  by  pass- 
ing through  a  sheet  of  yellow  glass,  we  shall  find  that  the 
modification  consists  in  a  relatively  great  absorption  of  the 
blue  and  violet  lights,  and  that  it  transmits  the  remaining 
colors  of  the  spectrum  quite  completely.  A  similar  test  of 
a  blue  shows  that  it  very  strongly  absorbs  red,  orange,  and 
yellow,  but  is  quite  transparent  to  the  remaining  colors.  If 
one  of  these  colored  glasses  is  placed  over  the  other,  and  ordi- 
nary white  light  is  directed  through  both,  the  transmitted 
light  will  be  green,  this  being  the  only  color  to  which  both 
are  transparent.  This  is  very  like  the  experiment  which  the 
painter  makes  when  he  mixes  a  yellow  with  a  blue  pigment 
and  spreads  the  mixture  upon  a  surface.  From  such  a 
painted  surface,  light  would  come  to  the  eye  which  may  be 
traced  to  four  sources :  First,  that  which  is  reflected  directly 
from  the  outer  surfaces  of  the  pigment  grains ;  second,  that 
which  has  been  modified  by  absorption  through  yellow  par- 
ticles alone ;  third,  that  which  has  been  affected  by  the  blue 
particles  and  not  the  others;  and,  finally,  that  which  has 
suffered  successive  absorption  by  both  yellow  and  blue  grains. 
Of  these  four  varieties  of  light  the  first  is  white,  the  second 
and  third,  being  complementary,  combine  to  form  white,  and 
the  fourth  is  left  to  determine  the  hue,  which,  as  has  been 
shown,  is  green. 

When  the  intensity  of  a  colored  light  is  greatly  increased, 
the  resulting  sensation  always  diminishes  in  saturation.  This 
fact  is  found  from  observation,  and  is  hardly  an  immediate 


176  LIGHT 

consequence  from  the  theory,  although  a  plausible  explana- 
tion might  be  deduced  from  it.  On  the  other  hand,  should 
the  brightness  of  a  colored  light  be  continuously  decreased, 
it  is  found  that  the  luminosity  falls  off  at  a  rate  differing 
greatly  with  different  colors.  Red,  for  example,  loses  its 
brightness  with  faint  illumination  very  rapidly,  whereas  blue 
lies  at  the  opposite  extreme  in  this  particular.  In  this  physi- 
ological fact  is  found  the  explanation  of  numerous  familiar 
phenomena.  We  may  cite  the  prevalent  bluish  tone  of  a 
moonlit  landscape  as  an  example ;  also  the  surprisingly  dark 
brown  color  of  an  orange  in  fading  daylight. 

It  has  been  already  stated  that,  as  far  as  purely  physical 
considerations  are  concerned,  any  three  colors  which  may  be 
so  combined  as  to  yield  white  might  be  made  the  basis  of  a 
color  diagram.  The  particular  set  chosen  by  Young  has  the 
advantage  of  giving  sensation  curves  which  have  but  a  single 
maximum  (see  Figure  44),  but  the  decisive  reason  for  choos- 
ing them  is  found  in  the  peculiarities  of  vision  exhibited  by 
color-blind  persons.  If,  for  example,  we  suppose  either  one 
or  two  of  the  three  fundamental  color-sense  organs  to  be 
defective  or  wholly  wanting,  a  perfect  theory  ought  to  suffice 
to  explain  the  differences  between  the  color  sensations  of 
such  a  defective  eye  and  those  of  the  normal  eye.  The  gen- 
eral success  attending  the  application  of  the  theory  as  stated, 
which  would  be  quite  lacking  in  another  essentially  different 
assumption  for  the  fundamental  colors,  is  a  strong  support  in 
favor  of  the  approximate  truth  of  the  theory. 

We  deduce  from  the  theory  that  an  eye  incapable  of  recog- 
nizing a  red  sensation  ought  to  find  matches  for  all  its 
color  sensations  in  a  proper  admixture  of  green  and  blue, 
diluted  with  white  or  darkened  if  necessary.  So,  too,  an  eye 
wholly  insensitive  to  green  ought  to  be  satisfied  with  a  linear 
color  diagram  containing  red  and  blue  only.  To  the  former 
of  these  defective  eyes  blue-green,  being  complementary  to 
red,  ought  to  appear  indistinguishable  from  white,  or  at  least 
from  the  white  of  low  luminosity  which  we  call  gray.  In 
exactly  the  same  way  we  infer  that  the  other  defect  would 


COLOR-BLINDNESS  177 

carry  with  it  the  inability  to  distinguish  a  difference  between 
magenta  (the  complementary  color  of  green)  and  gray.  These 
are  only  special  cases  among  a  large  number  of  predictable 
phenomena  from  the  theory,  but  they  will  serve  our  imme- 
diate purpose.  Such  defective  color  sense  is  far  from  uncom- 
mon, at  least  among  men  of  whom  we  may  perhaps  reckon 
four  in  every  hundred  as  more  or  less  red-blind;  green- 
blindness  is  far  less  frequent,  and  blue -blindness,  although 
not  wholly  unknown,  is  of  extreme  rarity.  Color-blindness 
is  relatively  extremely  uncommon  among  women. 

It  is  often  desirable  to  determine  the  character  of  color 
vision  in  persons  who  are  quite  uneducated  and  who  do  not 
know  the  names  of  more  than  three  or  four  colors.     Holm- 
gren has  devised  a  remarkably  simple  and  effective  method 
of  making  such  determinations,  wholty  independent  of  the 
power  possessed  by  the  individual  to  name  colors  correctly, 
and  founded  upon  the  theory  which  has  occupied  our  atten- 
tion.    It  may  be  easily  described  and  easily  applied.     The 
examiner  provides  himself  with  a  large  number  of  skeins  of 
worsted   representing  a  wide  range  of  colors,  not  only  as 
regards  hues,  but  also  in  respect  to  saturation  and  luminosity. 
One  of  these  skeins  is  of  a  magenta  color,  and  it  is  at  first 
kept  hidden  from  the  observer.     The  person  to  be  tested  is 
shown  a  skein  of  green  worsted,  and  is  requested  to  select 
from  the  purposely  confused  collection  all  those  which  have 
the  same  color,  irrespective  of  likeness  in  regard  to  saturation 
and  luminosity.     If  the  task  is  performed  without  error,  the 
evidence  of  normal  sense  is  almost  absolute ;  but  if  grays  or 
browns  or  other  colors  are  selected  as  belonging  to  the  same 
class  as  the  green,  abnormal  color  sense,  or  color-blindness, 
is  demonstrated.     It  remains  to  find  the  character  of   the 
defect.     In  the  first  place,  one  may  assume  that  it  cannot 
be   blue-blindness   on   account   of   its  extreme   rarity;   it  is 
therefore  either  red-  or  green-blindness.     The  next  step  in 
the   test  is   to   exhibit   the  skein  of  magenta  worsted,  and 
request  that  all  skeins  of  its  color  should  be  chosen.     If  the 
eye  examined  is  blind  to  red,  only  the  blue  present  in  the 

12 


178  LIGHT 

magenta  produces  a  color  sensation ;  consequently  violets  and 
blues  will  be  selected  as  allied  in  color  to  the  magenta.  If, 
on  the  other  hand,  the  color-blind  person  finds  in  grays  a 
match  for  the  magenta,  he  betrays  the  possession  of  green- 
blindness,  since  the  magenta  is  complementary  to  green. 

When  the  consequences  of  the  theory  are  pursued  to  the 
extreme,  however,  especially  in  the  case  of  the  faintest  visual 
perceptions,  it  proves  less  satisfactory.  For  instance,  if  in 
a  perfectly  dark  room  one  observes  a  piece  of  iron  under 
continuously  rising  temperature,  the  first  visual  impressions 
ought  to  be,  according  to  the  theory,  those  of  redness, 
since  the  first  copious  radiations  are  of  long  wavelength ;  but 
experience  shows  that  the  iron  becomes  visible  long  before 
any  trace  of  color  can  be  detected.  Another  fact  bearing 
upon  this  point  is  that  even  the  normal  eye  is  color-blind  for 
all  objects  very  remote  from  the  centre  of  most  distinct 
vision.  The  only  reason  for  this  singular  fact  being  wholly 
overlooked  is  because  few  people  pay  any  careful  attention 
to  the  character  of  oblique  vision.  Light  sensations  do  not, 
therefore,  necessarily  involve  a  sensation  of  color.  A  pecu- 
liarity of  the  construction  of  the  retina  has  suggested  an 
addition  to  the  theory.  It  is  found  that  the  cones  are  most 
abundant  in  the  macula  fovea,  or  region  of  most  distinct 
vision,  even  to  the  exclusion  of  the  rods  altogether,  while 
away  from  this  locality  the  rods  become  more  and  more 
abundant  up  to  the  limit  of  the  visual  field,  where  the  cones 
are  very  infrequent.  If  we  assume  that  the  seat  of  color 
sensation  is  in  the  cones,  while  a  stimulation  of  the  rods  gives 
rise  to  a  sense  of  brightness  only,  we  shall  have  a  theory 
somewhat  better  adapted  to  explain  known  phenomena,  and 
one  which  has  considerable  support  among  physiologists. 


CHAPTER   IX 
THEORIES   CONCERNING  THE  NATURE  OF   LIGHT 

ACCORDING  to  the  views  of  Newton,  visual  sensations 
were  produced  by  minute  corpuscles  projected  with  enormous 
velocities  from  luminous  bodies,  differences  of  color  being 
due  to  differing  size  in  these  minute  bodies.  When  these 
corpuscles  approach  the  boundary  of  an  optically  denser 
medium,  they  are  subjected  to  a  force  of  attraction  which 
causes  them  to  deviate  from  their  otherwise  rectilinear  paths. 
This  is  the  explanation  of  the  phenomenon  of  refraction.  The 
secondary  phenomenon  of  dispersion  was  simply  and  nat- 
urally explained  by  an  assumption  that  this  attracting  force 
varies  with  differing  size.  Singularly  enough,  the  explana- 
tion of  one  of  the  most  common  phenomena,  that  of  partial 
reflection  at  the  boundary  of  a  transparent  medium,  offered 
formidable  difficulties :  How  is  an  attraction  which  is  neces- 
sary to  account  for  refraction  also  to  act  as  a  repulsive 
force  in  the  case  of  a  portion  of  the  corpuscles  ?  This  is  a 
difficulty  which  the  advocates  of  Newton's  theory  have  never 
been  able  to  meet  in  a  satisfactory  manner. 

When  Newton  attempted  to  extend  his  theory  to  the 
explanation  of  the  colors  of  thin  plates,  a  subject  which  he 
was  the  first  to  investigate  in  a  scientific  manner,  it  was 
found  even  less  satisfactory.  He  was  obliged  to  supplement 
it  with  the  highly  artificial  hypothesis  that  the  corpuscles 
experience  periodic  changes  in  the  ease  with  which  they 
enter  a  refracting  body.  Even  this  addition  to  the  theory 
fails  to  yield  more  than  a  rough  approximation  to  an  expla- 
nation of  the  phenomena,  since  the  blackness  of  the  central 
spot  in  Newton's  rings  apparatus,  when  the  plates  are  brought 


180  LIGHT 

into  contact,  is  in  contradiction  with  it.  But  it  was  only  on 
account  of  a  subsequently  accumulated  knowledge  of  optical 
phenomena  which  refused  to  adjust  themselves  to  this  theory, 
no  matter  how  it  might  be  modified,  that  its  final  overthrow 
came  about.  This  not  only  required  a  long  time,  but  also  a 
champion  of  transcendent  power  to  break  with  a  system 
which  had  the  force  of  tradition  as  well  as  the  authority  of 
the  greatest  of  all  philosophers  to  support  it. 

From  1704,  the  date  of  the  publication  of  Newton's  Optics, 
until  1815,  the  corpuscular  theory  was  hardly  questioned ;  at 
any  rate,  it  reigned  supreme  in  all  treatises  on  light,  and  was 
questioned  only  by  a  very  few  investigators,  who  failed  to 
acquire  an  influence  that  was  anywhere  decisive.  In  the 
latter  year  a  remarkable  man,  Augustin  Fresnel,  a  young 
French  engineer  in  government  employ,  entered  upon  a 
career  of  scientific  activity  which  proved  of  almost  unprece- 
dented brilliancy  and  success.  This,  as  far  as  it  bears  upon 
the  purely  physical  theory  of  light,  may  be  regarded  as  com- 
pleted in  1826.  Beginning  with  a  highly  philosophical  criti- 
cism of  some  of  the  teachings  of  the  accepted  doctrines  of 
optics,  supported  by  the  most  apt  appeals  to  experiment, 
he  quickly  extended  his  investigations  until  they  embraced 
nearly  all  phenomena  of  light  known  to  his  contemporaries ; 
and  with  such  success  that  he  established  as  beyond  question 
the  essential  truth  of  a  wave  theory,  bringing  it  to  so  high  a 
degree  of  completion  that  his  views  were  long  supposed  to  be 
practically  final.  On  account  of  the  importance  of  this  work 
of  Fresnel  in  the  history  of  physical  science  of  the  past  cen- 
tury, it  is  well  worth  our  while  briefly  to  review  his 
achievements. 

The  phenomena  of  diffraction  first  engaged  the  attention 
of  Fresnel.  It  had  long  been  known  that  the  shadow  of  an 
opaque  body  cast  by  a  point-source  of  light  is  somewhat 
different  from  what  would  be  supposed  from  simple  geomet- 
rical considerations,  the  difference  consisting  chiefly  in  an 
encroachment  of  the  light  upon  the  borders  of  the  shadow. 
Newton,  who  called  this  phenomenon  inflection,  attributed  it 


THEORIES   CONCERNING   THE  NATURE   OF  LIGHT    181 

to  an  attractive  force  exerted  by  the  opaque  body  upon  the 
corpuscles  while  in  its  neighborhood,  thus  causing  an  in- 
bending  of  their  paths.  Fresnel  showed  that  this  theory  was 
quite  untenable,  since  the  inflection  caused  by  the  back  of  a 
razor  is  exactly  the  same  as  that  caused  by  the  edge,  although 
in  the  former  case  it  is  manifest  that  the  time  during  which  the 
corpuscles  are  subject  to  the  deflecting  force  is  far  greater 
than  in  the  latter.  By  similar  appeals  to  ingenious  crucial 
experiments  he  demonstrated  that  none  of  the  current  the- 
ories was  sound ;  but  far  from  resting  here,  he  showed  that 
all  the  observed  phenomena  could  be  perfectly  accounted  for 
in  the  undulatory  theory  of  light,  by  an  application  of  the 
principle  of  Huyghens.  Extending  this  principle,  so  fertile 
in  his  hands,  to  wider  fields  in  the  domain  of  optics,  he  found 
in  every  case  that  the  new  method  was  adequate  to  yielding 
perfectly  satisfactory  explanations.  Starting  with  quite  sim- 
ple and  natural  hypotheses  as  to  the  conditions  which  must 
exist  at  the  common  boundary  of  two  transparent  media, 
he  was  even  able  to  deduce  quantitative  laws  governing  the 
phenomena  of  reflection  and  refraction,  which  accord  surpris- 
ingly well  with  experiments  devised  to  test  them. 

A  few  years  before  the  commencement  of  Fresnel's  activ- 
ities, Malus,  while  looking  through  a  double-image  prism, 
observed  that  the  two  images  of  a  distant  window,  which 
happened  to  be  in  such  a  position  as  to  reflect  light  strongly 
to  his  eye,  were  quite  different  in  brightness,  and  under  some 
circumstances  one  image  might  entirely  disappear.  Further 
study  showed  that  all  ordinary  transparent  substances  were 
capable  of  thus  modifying  light  by  reflection,  and  that  the 
modification  is  complete  at  an  angle  which  is  simply  related 
to  the  refractive  index ;  moreover,  that  under  the  latter  con- 
ditions the  light  would  be  transmitted  through  a  doubly 
refracting  crystal  in  certain  directions  without  bifurcation. 
Such  modified  light  is  called  polarized  light,  and  the  phe- 
nomena described  are  two  of  the  simplest  of  an  enormously 
extensive  class,  many  of  which  are  of  extraordinary  beauty. 
This  discovery  and  those  which  quickly  followed  in  the 


182  LIGHT 

same  field  presented  a  host  of   new  and   difficult   problems 
to  physicists.     Of  the  many  active  and  able  workers  in  this 
domain  Fresnel  was  easily  the  leader.     In  a  very  few  years 
he   had  proposed  and  developed  a  general  theory  of   light 
which  embraced  these  new  phenomena  and  which  stood  almost 
unquestioned  until  our  own  day.     Although  this  subject  is 
far  too  extensive  and  intricate  for  any  adequate  presentation 
here,  we  may  consider  its  most  general  outlines.     Fresnel's 
theory  supposes  that  the  motion  of  the  particles  which  consti- 
tute the  vibrations  of  light  is  always  in  a  direction  at  right 
grngles  to  the  line  of  propagation  of  the  waves.     When  the 
paths  of  the  particles  are  quite  irregular  or  without  order, 
the  light  is  ordinary  light;  but  when  the  paths  are  similar, 
whether  straight  lines,  ellipses  with  their   axes  parallel,  or 
circles  with  a  common  direction  of  motion,  the  light  is  said 
to  be  polarized.     From  this  simple  hypothesis  he  succeeded 
in  erecting  an  extraordinary  structure  which  harmonized  and 
explained   nearly  every  known   phenomenon   of   light   in  a 
manner  that  until  the  most  recent  times  practically  withstood 
all  destructive  criticism.     Even  recent  achievements  in  this 
domain  of  science  have  been  supplementary  to,  rather  than 
subversive   of  Fresnel's  general  work.     Of   the  phenomena 
known  to  his  contemporaries,  that  of   dispersion  alone  was 
unconsidered  by  him,   a   phenomenon   which   obviously  de- 
pends upon  the  ultimate  molecular  structure  of  the  refract- 
ing  substance    and   which    has    recently  been    reduced    to 
comparatively  simple  laws.     This  great  work  of  Fresnel  was 
looked  upon,  as  indeed  it  well  deserves  to  be,  as  one  of  the 
greatest  monuments  to  the  human  understanding  —  compar- 
able to  Newton's  doctrine  of  universal  gravitation  —  and  it 
long  remained  of  almost  unquestioned  authority.     Ultimately, 
however,  one  of  its  fundamental  postulates,  namely,  that  the 
vibrations  are  always  at  right  angles  to  the  direction  of  the 
motion  of  the  light,  began  to  give  rise  to  difficulties.     The 
fact  also  that  the   theory  could   not   determine   specifically 
whether  the  direction  of  vibration  of  plane  polarized  light  is 
in  the  plane  of  polarization  or  perpendicular  to  it  was  not 


THEORIES   CONCERNING   THE  NATURE   OF  LIGHT    183 

only  a  manifest  incompleteness,  but  it  was  a  constant  stimu- 
lus to  a  critical  inspection  of  its  premises.  The  more  these 
points  were  studied  the  more  insoluble  the  difficulties  ap- 
peared, until  there  came  to  be  a  tolerably  wide -spread  belief 
that  the  theory  was  not  only  incomplete,  but  that  in  some  way 
it  must  be  essentially  in  error.  To  acquire  a  notion  of  what 
modern  science  has  done  to  clear  up  these  points,  we  must 
first  review  a  class  of  phenomena  which  seem  to  be  totally 
unconnected  with  optics,  but  which  in  the  end  will  afford 
a  very  remarkable  example  of  the  tendency  of  all  science 
toward  unity. 

In  1845  Faraday  discovered  that  if  polarized  light  is 
passed  through  a  transparent  substance  in  a  magnetic  field 
and  in  the  direction  of  the  field,  the  plane  of  polarization  is 
rotated.  The  amount  of  rotation  for  any  given  substance  is 
found  to  be  proportional  to  the  strength  of  the  magnetic  field 
and  to  the  length  of  the  path  in  the  material.  As  many 
substances,  such  as  turpentine,  a  solution  of  common  sugar, 
quartz  crystals  in  the  direction  of  their  crystalline  axes,  etc., 
present  us  with  a  similar  fact,  this  would  not  be  so  surpris- 
ing save  for  a  remarkable  difference  in  the  two  cases  which 
may  be  thus  described:  When  the  plane  of  polarization  is 
rotated  by  passing  through  a  sugar  solution  or  a  similar  body, 
and  the  transmitted  light  is  reflected  back  upon  its  course 
so  as  to  retrace  its  path,  it  is  found  that  the  original  angle 
of  the  polarization  is  perfectly  restored  by  a  precisely  equal 
rotation  in  the  opposite  direction  in  the  return;  but  a  similar 
experiment  upon  the  body  giving  the  magnetic  rotation  shows 
a  doubled  change  of  the  angle.  This  indicates  that,  although 
in  the  first  case  we  must  explain  the  rotation  by  the  molec- 
ular constitution  of  the  material,  we  are  not  permitted  to 
suppose  that  the  magnetic  field  has  produced  a  similar  molec- 
ular structure  in  the  second  case,  since  the  rotation  is  con- 
stant in  direction  irrespective  of  the  direction  of  the  motion 
of  the  light.  Of  course,  from  the  known  nature  of  magnet- 
ism, this  is  equivalent  to  asserting  that  there  must  be  some 
relation  between  light  and  electricity.  But  this  is  not  the 


184  LIGHT 

most  obvious  connection  between  these  two  classes  of  phe- 
nomena, for,  as  we  now  know,  the  earliest  division  of  mate- 
rials in  accordance  with  their  electrical  properties  involved 
a  classification  according  to  their  most  characteristic  optical 
properties  also.  Thus  all  conductors  of  electricity,  except- 
ing only  those  liquids  which  undergo  a  chemical  decomposi- 
tion when  they  transmit  an  electrical  current,  and  therefore 
belong  to  an  obviously  different  class,  are  extremely  opaque 
to  light;  conversely,  all  substances  which  are  good  insulators 
are  also  transparent  to  light,  at  least  to  an  extent  which 
would  make  a  sheet  a  few  hundred-thousandths  of  an  inch 
in  thickness  appear  perfectly  transparent,  although  such  a 
sheet  of  any  metal  or  similar  conductor  would  not  differ 
greatly  in  opacity  from  a  thick  plate.  An  excellent  illustra- 
tion of  the  generality  of  this  law  is  furnished  by  the  element 
carbon,  which  in  the  dense  opaque  form  —  like  graphite,  for 
example  —  is  a  very  good  conductor  of  electricity,  but  in  the 
form  of  the  transparent  diamond  is  an  insulator. 

Before  the  middle  of  the  past  century  two  methods  of 
measuring  electrical  magnitudes  had  been  developed;  one  of 
these  is  based  upon  the  repulsion  which  exists  between  two 
electrically  charged  bodies,  and  the  other  upon  the  repulsion 
which  exists  between  two  similar  magnet  poles.  Elaborate 
and  repeated  investigations  have  demonstrated  that  if  a  given 
electrical  magnitude  is  measured  according  to  one  of  these 
systems,  and  the  value  thus  found  is  compared  to  a  measure- 
ment of  the  same  quantity  in  the  other  system,  the  ratio 
involves  a  velocity  only.  This  statement  is  quite  independent 
of  the  kind  of  magnitude  chosen  for  the  experiment.  Within 
the  limit  imposed  by  unavoidable  errors  of  observations,  the 
value  of  this  velocity  always  appears  to  be  the  same  as  the 
velocity  of  light. 

Here,  therefore,  are  three  distinct  relations  between  light 
and  electricity,  which  have  been  long  known,  to  no  one 
of  which  is  it  possible  to  attach  any  a  priori  reason.  It 
was  left  to  Maxwell  to  illuminate  this  obscure  field. 
His  long  and  successful  investigations  in  electricity  and 


THEORIES   CONCERNING   THE  NATURE   OF  LIGHT    185 

magnetism,  especially  his  efforts  to  reduce  Faraday's  brilliant 
discoveries  to  correlation  and  to  consistent  scientific  state- 
ment, led  him  to  the  conclusion  that  light  itself  consists  of 
electrical  vibrations.  He  attempted  to  test  the  validity  of  this 
hypothesis  by  every  means  at  his  command.  For  example, 
according  to  his  theory  a  non-magnetic  substance  ought  to 
have  a  dielectric  constant,  or  what  Faraday  named  its  specific 
inductive  capacity,  proportional  to  the  square  of  its  index  of 
refraction.  This  indicated  relation  was  found  to  hold  with 
all  expected  precision  in  some  cases,  but  to  be  widely  re- 
moved from  the  truth  in  others.  Again,  since,  according  to 
the  theory,  only  those  substances  are  transparent  which  will 
offer  a  resistance  to  the  motion  of  electricity  within  them 
analogous  to  elastic  reaction,  there  ought  to  be  a  de termi- 
nable relation  between  electrical  conductivity  and  opacity. 
Maxwell  attempted  to  find  this  relation  in  the  case  of  gold- 
leaf,  which  is  sufficiently  thin  to  transmit  a  measurable  por- 
tion of  light  falling  upon  it.  Notwithstanding  that  the 
discrepancy  was  here  found  disappointingly  great,  the  grad- 
ual accumulation  of  knowledge  of  the  more  recondite  phe- 
nomena of  the  electrical  field  had  led  the  great  majority  of 
physicists  to  the  conclusion  that  Maxwell's  theory  was  at  least 
a  close  approximation  to  the  truth,  and  accordingly  one  of 
the  most  brilliant  discoveries  of  the  nineteenth  century. 
This  may  be  regarded  as  a  fair  statement  of  the  attitude  of 
the  world  of  science  in  1888,  when  Hertz,  a  German  physicist 
of  extraordinary  merit,  whose  early  death  was  a  great  loss  to 
science,  made  a  series  of  remarkable  experiments  which  have 
eliminated  all  possible  doubt  as  to  the  essential  verity  of 
Maxwell's  theory  of  light.  Fortunately  it  is  not  difficult  for 
us  to  gain  a  sufficient  knowledge  of  the  character  of  these 
experiments  to  enable  us  to  understand  their  general  bearing. 
It  had  long  been  known  that  a  Leyden  jar  suddenly  dis- 
charged through  a  thick  wire  gives  rise  to  an  oscillatory 
current  of  very  brief  duration,  and  that  in  certain  simple 
cases  the  period  of  the  oscillations  can  be  calculated  with 
considerable  precision.  Hertz  recognized  that  during  the 


186  LIGHT 

time  of  discharge  such  an  electric  circuit  must  be  a  source  of 
oscillatory  changes  in  the  magnetic  field,  which,  if  the  views 
of  Maxwell  are  in  accordance  with  fact,  should  be  propa- 
gated through  space  with  the  velocity  of  light.  Although 
it  is  difficult,  if  not  quite  impossible,  to  measure  directly  this 
velocity,  if  one  knows  the  wavelength  and  the  period  it  is 
perfectly  easy  to  deduce  the  velocity  from  these  two  ele- 
ments, since  in  its  period  every  wave  moves  a  distance  equal 
to  its  own  length.  In  these  experiments  the  period  was 
found  by  calculation  from  the  elements  of  the  electric  cir- 
cuit; it  only  remained  therefore  to  determine  the  length  of 
the  waves.  Hertz  accomplished  this  in  the  following  simple 
and  ingenious  manner :  At  a  considerable  distance  from  the 
source  of  the  waves  he  placed  a  large  sheet  of  metal  perpen- 
dicular to  its  direction  from  the  source.  From  this  sheet 
the  waves  were  returned  upon  themselves  by  reflection.  Now, 
a  well-known  fact  in  wave  motions  is  that  when  two  systems 
of  waves  of  like  period  are  moving  in  opposite  directions,  they 
combine  to  form  a  system  of  standing  waves  of  half  the 
length  of  the  free  waves.  The  regions  where  motion  is  de- 
stroyed by  this  kind  of  interference  are  called  nodes.  Hertz 
demonstrated  the  existence  and  positions  of  these  nodes  by 
means  of  an  apparatus  which  possessed  the  same  electrical 
period  as  his  source.  This  apparatus  he  called  a  resonator. 
The  value  of  the  velocity  of  these  waves  deduced  from  his 
observations  differs  no  more  from  the  known  velocity  of  light 
than  would  be  expected  from  the  unavoidable  errors  of  ob- 
servation; thus  it  complies  with  the  requirements  of  Max- 
well's theory.  These  waves,  therefore,  are  shown  to  differ 
from  light  waves  only  in  their  enormously  greater  wave- 
lengths, and  they  must  be  subject  to  all  the  established  laws 
of  optics  which  are  independent  of  the  length  of  the  waves. 
The  last  conclusion  was  thoroughly  tested  by  Hertz  by  a 
series  of  most  interesting  and  convincing  experiments.  He 
found  that  strictly  according  to  the  laws  of  optics  these  waves 
are  reflected  from  the  surfaces  of  all  bodies  which  conduct 
electricity ;  that  they  readily  pass  through  substances  which 


THEORIES   CONCERNING    THE  NATURE   OF  LIGHT    187 

behave  as  insulators;  and  that  in  passing  from  one  insulating 
medium  to  another  the  direction  of  propagation  is  altered 
in  accordance  with  the  law  of  sines.  Further  than  this,  he 
showed  that  such  electrical  waves  admit  of  polarization,  and 
they  are  therefore  characterized  by  motions  at  right  angles 
to  the  direction  of  propagation. 

During  the  time  which  has  elapsed  since  these  investiga- 
tions, a  host  of  experimenters  have  improved  the  methods 
and  apparatus  of  Hertz,  and  have  largely  extended  the  range 
of  wavelengths  that  can  be  observed.  At  present  these 
Hertzian  waves  are  being  applied  to  transmitting  electrical 
signals  to  distant  points  without  the  use  of  connecting 
wires.  On  the  other  hand,  many  investigators  have  been 
employed  in  the  application  of  analysis  to  both  the  old  and 
the  new  problems  in  optics.  The  difficulties  which  attach 
to  Fresnel's  mode  of  regarding  the  optical  phenomena  of 
crystalline  media  are  found  to  disappear,  and  all  the  complex 
phenomena  of  light  admit  of  explanation  from  a  consistent 
body  of  premises. 

This  great  advance,  however,  cannot  be  regarded  as  an 
unmixed  gain,  since  it  replaces  comparatively  simple  mechan- 
ical considerations,  from  which  we  can  construct  mental 
pictures  of  the  phenomena  in  a  medium  submitted  to  the 
action  of  radiant  energy  closely  allied  to  those  which  are 
suggested  by  ordinary  experience,  by  conceptions  of  agencies 
that  no  philosopher  has  yet  succeeded  in  grasping  even  to  an 
extent  necessary  to  form  a  clear  notion  of  what  constitutes 
an  electrically  charged  body;  much  less  what  an  electric 
vibration  may  be.  It  is  not  surprising  that  the  earlier  views 
are  the  simpler  ones,  else  they  hardly  would  have  had  pre- 
cedence in  time;  but  there  is  no  possible  doubt  that  no 
purely  mechanical  theory  has  been  formulated,  even  after  a 
century  of  effort,  which  is  proof  against  all  criticism,  nor, 
on  the  other  hand,  has  any  one  hitherto  been  able  to  urge 
an  objection  to  the  electrical  theory  now  prevalent,  save  the 
difficulty  of  understanding  its  fundamental  conceptions.  We 
may  hope  to  have  some  time,  perhaps  at  no  very  distant 


188  LIGHT 

future,  a  popular  treatise  on  light  which  shall  start  from 
the  phenomena  presented  by  a  rubbed  stick  of  sealing-wax, 
and  by  logical  development  deduce  all  the  phenomena  of 
light  now  known,  and  very  likely  a  host  of  others  not  yet 
dreamed  of. 


APPENDIX  A 

THE  usual  formulas  for  the  calculations  of  lens  systems  other 
than  those  of  the  greatest  simplicity  are  extremely  complicated. 
Should  such  accuracy  be  desired  as  is  absolutely  requisite  for 
the  designing  of  optical  apparatus,  in  which  case  the  ordinary 
approximations  are  insufficient,  the  formulas  become  almost 
unmanageable. 

The  famous  paper  by  Gauss,  which  was  published  in  1840,  and 
which  first  introduced  the  conception  of  cardinal  points  in  an 
optical  system,  was  a  very  remarkable  advance  on  the  older  theo- 
ries. After  its  appearance  it  was  possible  to  take  into  account 
the  thicknesses  and  separations  of  the  lenses  with  rigid  accuracy, 
without  introducing  greater  complexity  in  the  final  equations  than 
were  already  secured  by  a  method  which  assumed  that  these  ele- 
ments might  be  neglected.  But  even  more  important  than  this 
was  his  perfect  definition  of  terms  which  had  been  used  for  ideal 
systems,  but  which  could  not  have  any  exact  meaning  when  ap- 
plied to  any  physical  apparatus,  such  as  focal  length,  magnifica- 
tion, optical  centre.  This  paper,  with  Listing's  addition  of  the  two 
nodal  points,  left  nothing  to  be  desired  in  giving  a  complete  and 
definite  geometrical  notion  of  an  optical  system  whose  cardinal 
points  were  known.  But  this  is  not  the  primary  end  either  in 
designing  an  optical  system  or  in  determining  its  optical  effi- 
ciency when  completed.  The  absolute  focal  length  and  mag- 
nification of  any  optical  instrument  is  of  the  least  possible 
consequence,  as  becomes  at  once  obvious  when  we  consider  how 
few  working  astronomers  or  microscopists  can  give  these  data 
with  precision  for  their  own  instruments. 

What  is  necessary  in  designing  an  optical  instrument  is  a 
knowledge  of  its  aperture,  in  the  more  general  sense,  of  the 
variation  of  its  focal  plane  with  varying  wavelength  of  light,  of 
its  spherical  aberration,  of  the  variation  of  the  spherical  aberra- 


190  LIGHT 

tion  with  varying  wavelength,  and,  if  the  character  of  the  image 
remote  from  the  axis  is  of  consequence,  the  variation  of  the 
magnification  with  varying  wavelength,  and,  finally,  its  astigma- 
tism. Neither  the  older  methods  nor  that  of  Gauss  defines  the 
first  of  these  important  quantities,  except,  of  course,  in  the  sim- 
ple case  of  the  telescope,  nor  are  they  convenient  for  calculating 
the  others. 

From  the  character  of  the  quantities  named  above,  it  is  obvious 
that  their  mathematical  expressions  must  be  for  the  most  part  of 
the  nature  of  derivations  from  the  fundamental  equations.  But 
all  the  fundamental  equations  for  lenses  of  sufficient  exactness  to 
be  of  the  least  value  are  so  complicated  that  their  derivations,  at 
least,  are  quite  valueless  for  practical  purposes. 


P.       f 


FIGURE  47. 

The  use  of  the  natural  conceptions  of  wave-surfaces,  of  refract- 
ing surfaces  characterized  by  their  curvatures,  and  of  light  ve- 
locities, which  has  been  employed  in  the  text  of  this  book,  may 
well  possess  a  commensurate  advantage  when  reduced  to  rigidly 
quantitative  form.  Indeed,  I  have  myself  been  well  content  with 
this  innovation  during  many  years  of  use  in  practice.  The  fol- 
lowing sections  may  be  looked  upon  as  a  brief  exposition  of  the 
efficiency  of  the  method  and  as  a  proof  of  all  the  principles 
employed  in  the  text.  It  should  be  added  that  nearly  all  of 
this  appendix  is  a  slightly  modified  reproduction  of  a  paper  in 
the  Memoirs  of  the  National  Academy  for  1893. 


APPENDIX  A  191 


I.    GENERAL    EQUATIONS 

Let  p,  pf,  and  p\  be,  respectively,  the  geometrical  centres  of  the 
lens  surface,  the  incident  wave-surface,  and  the  refracted  wave- 
surface  ;  also  let  y,  c',  and  cl  be  the  curvatures  of  these  surfaces 
in  order,  a  positive  curvature  corresponding  to  the  case  where  the 
centre  of  curvature  is  in  the  direction  of  propagation  of  light 
from  the  surface.  Let  x,  x',  and  x±  be  the  sagittas  of  these  sur- 
faces in  order  named,  corresponding  to  the  common  semi-chord 
y  ;  then  for  small  values  of  y  we  have 


From  the  laws  of  wave  propagation  we  have,  if  p  represents 
the  ratio  of  the  velocity  in  the  medium  to  the  right  of  the  sur- 
face y  to  that  in  the  medium  to  the  left, 

x  -  a?!  =  p  (x  -  xf)  . 

Substituting  the  above  values  of  x}  x1,  Xi  in  this  equation,  elim- 
inating the  common  factor  £  y2,  and  solving  for  c1?  we  have 

Ci  =  y  (1  -  P)  +  Pc'. 

This  wave-surface  will  be  propagated  with  uniform  velocity 
and  uniformly  decreasing  radius  until  it  reaches  the  second  re- 
fracting surface  at  a  distance  t  from  the  first,  when  its  radius 

is  reduced  from  —to  -  —  £,  and  therefore  its  curvature  to  the 

Cl  C\ 

reciprocal  of  this,  or  to 


1-arf 

if  we  define  ^  by  this  equation.  After  refraction  at  this  second 
surface,  the  curvature  will  be  according  to  exactly  the  same  rea- 
soning as  that  applied  to  the  first  surface, 

i 

This  completes  the  general  solution,  and  it  can  be  extended  to 
any  number  of  refractions,  thus :  — 


192  LIGHT 


ci  =  y  (1  -  p)  +  W'  ^  =  1 

c2  =  y  (1  -  p')  +  JI'P'C!  f'  =  (1  -  erfi)-1 

cs  =  y  (1  -  p")  +  /'  p"c2  ^/  =  (1  _  (c2t2)  -1     (a) 


/  (1  ~  PA)  +  ^  /  ^A  pA  -  (1  -  CA  «A)-i 

If  the  lenses  are  all  indefinitely  thin  and  in  contact  —  in  other 
words,  if  all  the  tf's  are  equal  to  zero,  and  therefore  all  the  ^u's 
equal  to  unity  —  the  equations  (a)  can  be  readily  combined  into 
one.  This  reduction,  by  inspection,  is 


cA+1  =  /  (1  -  f)  +  /-1  (1  -  p*-1)  p*  + 


If  n0  is  the  index  of  refraction  of  the  first  medium,  nf  in  that 
of  the  second,  and  so  on,  we  shall  have 

_  ?i°       _  n1  n* 

P~^>P    "^  p   '"-rt**' 

and  the  product  of  all  these  quantities  will  be  equal  to  -j 


which  quantity,  the  ratio  of  light  velocity  in  the  last  medium  to 
that  in  the  first,  we  will  designate  by  p0.  By  this  substitution 
the  above  equation  becomes 


where  P  is  a  constant,  depending  solely  upon  the  physical  con- 
stants of  the  system ;  it  is  called  the  power  of  the  combination, 
not  only  because  it  is  the  change  in  curvature  which  the  system 
can  produce  in  a  plane  incident  wave-surface  (cf  =  0),  but  also 
because  in  the  most  common  of  all  cases  in  which  the  first  and 
last  media  are  alike,  and  therefore  p0=l,  P  is  the  curvature 
which  the  system  adds  to  every  incident  wave-surface. 

II.     IMAGES    AND    MAGNIFICATION 

Let  of  (Figure  48)  be  the  indefinitely  small  distance  of  the  centre 
of  the  incident  wave-surface  cf  from  the  axis,  and  ol  that  of  the 
centre  of  the  refracted  wave-surface  cx ;  then  we  have,  from  the 


APPENDIX  A 


193 


general  law  of  wave  motion,  the  line  o^  is  the  image  of  o',  and 
the  angle  which  cx  makes  with  y  at  the  vertex  is  p  times  that 
which  c 'makes  at  the  same  point;  hence 


FIGURE  48. 


The  image  of  0X  after  refraction  at  a  second  surface  we  may  call 
02  ;  then  by  similar  reasoning  we  have 

°z  _    >  P'CI 

Oi  C2 

Extending  this  process  to  X  -f-  1  refractions  and  collecting,  we 
have 


2  __. 


A— 1, 


The  product  of  these  equations  is,  remembering  that  the  prod- 
uct of  all  the  p's  is  po, 


The  value  of  the  factor  ^\   is  easily  changed  by  substituting 
successively  the  value  of  p*  and  of  CA  given  in  (a),  to  the  form 

13 


194  LIGHT 


u^u*-1 
I     _ 


in  which  ax—  i  and  /?A_  i  are  constants  depending  solely  upon  the 
physical  constants  of  the  system,  and  not  at  all  upon  the  value  of 
any  of  the  wave  curvatures.  If  we  make  another  substitution  of 
the  values  of  ^—  1  and  of  c\—  l  in  this  last  equation,  we  have 


—   .A-2 


aA- 


If  we  repeat  this  process  of  substitution  A.  times,  we  shall  have 
finally, 

1 


in  which  a  and  /?  are  determinable  functions  of  the  y's,  the  t'  s,  and 
the  p's,  that  is,  of  the  physical  constants  of  the  system,  and  the 
ratio  of  the  ultimate  image  to  the  object,  or  the  magnification 
becomes 


There  is,  however,  another  very  important  expression  for  the 
magnification  which  may  be  derived  as  follows  :  — 

Suppose  we  have  a  diaphragm  at  the  first  surface  with  the 
small  diameter  2a.  Let  the  semi-angular  diameters  of  the  wave- 
surfaces,  incident  and  refracted,  thus  limited  be  designated  by  w' 
and  o>1?  and  so  on  for  the  successive  refracting  surfaces  ;  then  — 

sin  to'  =  acf 
sin  o>i  =  aci 
sin  MI  =  ctif^'Ci 
sin  0)2  = 
sin  o>2  = 
sin  w3  = 


sn  w\  = 
sin 


From  these  we  have 


APPENDIX  A 


195 


sin  a/ 
sin  <oj 
sin  tUj 
sin  w2 
sin  w 


_ 


//   2 


sin  WA 

— =-=  (A"  —  — 

sin  a>!  cs  sin  w  A-f-i 

The  product  of  these  equations  gives 
sin  ID'  c' 


sn 


Multiplying  this  equation  by  p0  and  observing  (b)  we  have 


sn 


o'  sin 

This  equation  has  been  proved  for  small  values  of  a  only,  and 
would  therefore  hold  good  if  either  o>  or  tg  <o  were  substituted  for 
sin  w.  In  the  last  form,  tg  o>  replacing  sin  w,  the  equation  was 
first  given  by  Lagrange,  and  was  a  most  important  contribution 
to  the  theory  of  optics ;  but  it  is  possible  to  show  that  the  form 


which  we  have  given  above  is  rigidly  true,  independently  of  the 
size  of  a,  provided  only  that  the  incident  and  finally  refracted 
wave-surfaces  are  both  spherical,  that  is,  that  the  system  is  free 
from  spherical  aberration. 


196  LIGHT 

"Letpq  (Figure  49)  be  the  incident  wave  cr,  and  prqr  the  finally 
refracted  wave-surface  c^.  If  c'  is  limited  by  a  diaphragm  at  q, 
then  c\+i  is  also  limited  at  a  point  qf,  which  is  the  corresponding 
point  to  q  ;  that  is  to  say,  the  point  in  c^-j-i  where  all  the  light 
energy  comes  from  a  region  in  cf  indefinitely  near  q.  Call  the 
semi-angular  aperture  of  cf  and  of  CA+I,  <o'  and  WA+I,  as  indicated 
in  the  figure.  Now  suppose  the  incident  wave  to  be  inclined  by 
an  indefinitely  small  angle  o'c',  then  the  finally  refracted  wave 
will  also  be  inclined  by  an  infinitely  small  angle  at  p',  which  will 
be  equal  to  0A_|_i  CA+I.  It  is  apparent  from  the  figure  that  in  the 
new  oblique  waves  qi  is  a  corresponding  point  to  q,  since  the  wave- 
surface  is  propagated  in  the  direction  of  its  normal,  and,  for  the 
same  reason,  q^  is  a  corresponding  point  to  qr,  hence  <£/  on  the 
finally  refracted  oblique  wave  is  a  corresponding  point  to  q±.  But 
since  p  and  p'  are  corresponding  points  for  both  wave  systems, 
the  time  required  for  light  waves  to  pass  from  q  to  qf  and  from 
ql  to  <?/  is,  in  each  case,  equal  to  the  time  required  in  going  from 
p  to  p'  ;  hence  the  time  required  for  progression  from  q  to  ql  is 
equal  to  the  time  from  qf  to  <?/.  The  velocity  of  propagation  in 
the  last  medium,  however,  is  p0  times  as  great  as  in  the  first  ; 
consequently  we  have 


By  inspection  of  the  figure  we  see  that 
qq^  =  pq  •  orcr  cos  ^  o/ 

q'q'i  =  p'q1'  °\+i  CM-I  cos 

Combining  these  three  equations  and  substituting  the  trigono- 
metrical expression  for  the  chords  pq  and  p'qf,  we  have 

2p0  of  sin  £  «/  cos  £  o>'  =  2  OA+I  sin  \  eox_|_i  cos  £  o>A_|_i  , 

whence  we  derive  immediately  the  equation  (c). 

This  highly  important  relation  is  essential  in  calculating  the 
absolute  optical  power  of  every  optical  apparatus  except  the  tele- 
scope. Its  truth  was  assumed  by  Professor  Abbe  in  his  cele- 
brated paper  on  the  limit  of  power  in  microscopes,  and  was 
proved  very  indirectly  by  Helmholtz  in  his  paper  on  the  same 
subject  as  a  consequence  of  the  second  law  of  thermodynamics. 
I  am  not  aware  that  any  proof  purely  optical  has  heretofore  been 
given. 


APPENDIX  A  197 

III.     SIMPLIFICATION    OP    GENERAL   EQUATIONS  (a) 

In  the  discussion  to  this  point  we  have  made  cr  the  curvature  of 
the  incident  waves  at  the  first  vertex  of  the  system,  and  CA+I  the 
curvature  of  the  finally  refracted  waves  at  the  last  vertex.  We 
will  now  seek  the  relation  of  the  curvatures  for  incident,  and 
finally  refracted  wave-surfaces  at  points  situated,  respectively,  at 
a  distance  x0  from  the  first  vertex  and  x1  from  the  last. 

Let  us  suppose  that  the  incident  wave-surfaces  are  bounded 
by  a  circular  diaphragm  of  radius  a0  at  XQ)  then  the  finally  re- 
fracted wave-surfaces  will  be  also  bounded  and  will  have  a  defi- 
nite semi-diameter  at  x^  which  we  shall  designate  by  a°i ;  then, 
if  C1  and  Ci  represent  the  wave  curvatures  at  the  points  x0  and  Xi 
for  incident  and  finally  refracted  waves,  respectively,  we  have 
the  following  relations  :  — 

sin  o>'  =  a°C"  c'  = 


1  +  C'x 
sin 


0 

f 

1 


Substituting  these  values  in  (c),  we  have 

0'  a°i  Ci 

By  replacing  the  left  member  of  this  equation  by  its  value  given 
in  (6)  and  then  substituting  the  values  above  for  c'  and  CA+I  we 
have 

a°  1  +  CiXi 


whence  the  value  of  d  is  given  by  the  equation 


If  x0  and  xl  are  so  chosen  that  one  is  the  optical  image  of  the 

other,  —  becomes  a  constant  since  a°i  is  simply  the  image  of 
a\ 

a°,  and  the  expression  for   Ci  becomes  greatly  simplified.     We 


198  LIGHT 

will  call  the  value  of  ^-  for  any  such  case  k.     This  value  can 
a  i 

be  found  by  substituting  —  ^  and  —  for  c'  and  CA+J  in  equation 

X  Xi 

(5),  whence,  remembering  that  a°  corresponds  to  o',  we  have 

1 


When  C1  =  0  the  value  of  6\  is    a~     ;  but  with  this  value  of 


C'  we  have  likewise  c'  =  0,  since  both  equations  mean  that  the  in- 
cident wave-surface  is  flat.  Compute  the  value  of  CA+I  for  this 
value  of  c'  by  means  of  equations  (a)  and  call  it  cVf  i  ;  then  from 
the  equation  above  connecting  CA+I  and  Cv  namely, 


we  find  ka-  I  =  kaco      > 

xl 

Substituting  these  expressions  in  the  general  equation,  we  have 
this  remarkably  simple  expression  to  replace  equations  (a), 

C^kaO\+l  +  p^Cf.  (d) 

Before  discussing  the  methods  for  finding  the  contents,  &,  a, 
and  cVfi  in  this  general  equation,  we  will  first  establish  expres- 
sions for  the  magnification  and  for  the  change  in  direction  of  the 
wave-surface  in  progressing  from  x0  to  x±. 

The  term  magnification  has  two  distinct  meanings,  namely,  the 
ratio  of  the  dimension  of  the  images  to  that  of  the  object  meas- 
ured at  right  angles  to  the  axis,  and  a  corresponding  ratio  in  the 
direction  of  the  axis.  The  first  we  may  call  the  transverse  mag- 
nification and  designate  M  ;  the  other  may  be  called  the  longitu- 
dinal magnification  and  designated  L. 

From  the  fifth  equation  of  this  section  we  may  write  at  once 
the  value  for  the  first  species  of  magnification.  It  is  equal  to 

*•=/>.*.  («) 


APPENDIX  A  199 

The  longitudinal  magnification  is  obviously  equal  to  the  ratio  of 
the  displacement  of  the  image  along  the  axis  to  the  correspond- 
ing displacement  of  the  object.  From  this  definition  and  equa- 
tion (d)  we  find 


C> 
From  these  two  equations  we  find 


Po 

This  law  explains  why  the  depth  of  field  in  microscopic  vision 
seems  so  small,  and  also  why  an  object  under  the  microscope  ap- 
pears so  much  flatter  when  mounted  in  a  medium  of  high  refrac- 
tive power,  for  in  this  case  p0  is  greater  than  unity. 

A  consideration  which  is  of  much  importance  is  the  relation  of 
the  directions  of  the  wave-surfaces.  This  relation  is  readily  de- 
termined for  the  points  x0  and  x^.  In  Figure  49  letp  andy  be  the 
points  x0  and  x^  respectively  ;  then,  since  by  definition  pf  corre- 
sponds to  p,  we  need  make  no  restrictions  as  to  the  value  of  the 
angles  of  inclination  of  the  oblique  wave-surfaces.  Call  these 
angles  </>0  and  fa  respectively  ;  then,  if  pq  is  small,  we  have,  as 
before, 


and 

Wi  =  P3  sin  < 
q'q\=p'q'  sin 

and  finally,  since  p'q'  is  the  image  of  pq, 


From  these  equations  we  read  at  once 

sm<^i_fc 
sin_-.*Po. 


200  LIGHT 


IV.      TO    FIND    THE   VALUES    OF    THE   CONSTANTS    IN 
EQUATION  (d) 

There  are  two  cases  presented  in  practice  :  First,  when  all  the 
constants  of  the  optical  system  are  given,  and,  second,  when  we 
can  only  depend  upon  measurements  as  applied  to  the  system  as 
a  whole.  We  shall  consider  these  two  cases  in  turn. 

Case  1.  —  All  the  constants  of  the  system  being  known,  com- 
pute COA+I  by  making  c'  equal  to  zero  in  equations  (a)  ;  the  value 
of  a  is  the  reciprocal  of  p\\  in  this  computation. 

To  find  k  we  either  assume  the  value  of  x0  and  thence  compute 
x1  and  kj  or,  assuming  the  value  of  k,  compute  x0  and  x^ 

For  the  first  method  make  c'  =  —  in  (a)  ;  the  resulting  value 

•     #o 

of  CA+I  equals  _  ,  and,  substituting  these  values  of  cf  and 


we 


in  (£),  remembering  that  (a  +  fid)  is  the  reciprocal  of  J*A_ 
find  the   reciprocal   of  k  at   once.     The   solution   is   therefore 
complete. 

If  k  is  assumed,  we  compute  c°A+i  and  a  as  before,  then 


whence  we  find  x0  from  (a)  by  making  CA+I  =  —  when  x0  =  -  . 

Xi  C' 

Case  2.  —  Should  we  desire  to  find  the  constants  of  (d)  by  ex- 
periment, proceed  as  follows  :  — 

Choose  any  convenient  point  in  the  axis  of  the  system  for  #0, 
placing  there  an  object,  for  example,  a  scale  of  equal  parts  ;  the 
image  of  this  object  will  be  at  x^  and  the  ratio  of  the  size  of  the 
object  to  the  image  will  be  k.  Then  find  the  place  on  the  axis  of 
the  image  of  an  object  at  an  indefinitely  great  distance  in  front  of 
the  system,  that  is,  when  <7'=  0.  The  reciprocal  of  the  distance 
of  this  point  from  Xi  equals  kac^+i,  and  the  problem  is  solved. 


V.       ON    PARTICULAR   VALUES    OF    k    IN    EQUATION    (d) 

Inspection  of  the  equation  (d)  suggests  at  least  four  values  of 
k,  which  make  the  equation  of  special  simplicity.     These  values 


APPENDIX  A  201 

are  1,  —  1,  —  ,  and         .     Substituting  these  in  turn,  and  replacing 

Po         —  po 
the  diacritical  marks  of  that  equation  by  convenient  symbols,  we 

have 


Cp  = 

C-p  =  -cA+1  +  poC-P  (d>1) 

p0Cn  =  aC 
p0C_«  -  - 

The  points  defined  by  x0  and  xj  when  k  =  1  are  called  the  first 
and  second  principal  points,  respectively.  They  were  first  intro- 
duced and  their  properties  defined  by  Gauss,  in  a  celebrated  paper 
published  in  1840.  We  see  at  once  that  the  image  of  a  small 
object  at  XQ  is  at  xi9  the  transverse  magnification  is  1,  and  the  lon- 

gitudinal magnification  is,  from  (/),  —  .     The  inclination  of  the 

Po 

incident  wave  at  x0  and  the  finally  refracted  wave  at  Xi  is  given 
by  (g),  which  becomes 


sin  <f>0 


_ 


For  k  =  —  1,  the  second  of  the  above  equations,  the  points  x0 
and  Xi  are  called  the  first  and  second  negative  principal  points. 
An  object  at  x0  has  its  image  at  ic1?  the  image  being  inverted,  but 
of  the  same  transverse  dimensions  as  the  object  ;  the  longitudinal 

magnification  is  the  same  as  before,  namely,  _  .   Finally  (g)  gives 

Po 

sin  <£j  _ 
sin  <£0 

For  k  =  —  ,  the  two  points  x0  and  Xi  are  called  the  nodal  points. 

Po 

These  were  first  investigated  and  named  by  Listing  in  1851.  An 
object  at  the  first  nodal  point  has  its  image  at  the  second  nodal 
point,  both  axial  and  transverse  magnifications  being  equal  to  p0. 
These  are  the  only  two  points  so  related  that  the  image  of  a  body 
at  one  point  has  the  same  shape  and  orientation  as  the  body  itself. 
A  more  important  property  is  derived  from  equation  (g),  in  which 
we  see  for  this  case 

sin  <#>i  _  -^  . 

sin  <~ 


202  LIGHT 

that  is,  a  wave-surface  which  would  pass  through  the  first  nodal 
point  at  an  inclination  <£0  passes  through  the  second  nodal  point 
under  the  same  inclination  after  final  refraction. 

The  final  form,  in  which  the  points  x0  and  Xi  may  be  called  the 
negative  nodal  points,  has  the  same  longitudinal  magnification 
for  these  points,  but  the  transverse  magnification  and  the  relation 
of  the  inclinations  are  equal  to  those  of  (diil)  taken  negatively. 

It  will  be  observed  that  (dl)  is  of  exactly  the  same  form  as  the 
equations  for  a  system  of  infinitely  thin  lenses  in  contact.  More- 
over, if  the  first  and  last  media  are  alike  —  in  which  case 
p0  =  1  —  equations  (dl)  and  (cZm)  become  identical,  as  do  also  (dli) 
and  (cZiv),  or,  in  words,  the  principal  and  nodal  points  fall  together, 
and  also  the  negative  principal  and  negative  nodal  points. 

The  determination  of  the  position  of  all  these  points  is  the 
problem,  when  the  constants  of 'the  system,  are  known,  of  Case  1 
of  the  preceding  section,  and  therefore  need  not  be  further  dis- 
cussed. But  to  determine  them  experimentally  is  not  the  same 
as  Case  2,  because  they  assume  determinate  values  for  k.  We 
may  proceed  as  follows  :  — 

If  both  principal  points  are  outside  of  the  system  and  on  oppo- 
site sides,  we  may  find  them  at  once  by  seeking  the  places  of  ob- 
ject and  image  when  the  image  is  erect  and  equal  in  size  to  the 
object.  This  process  is  applicable  to  some  forms  of  compound 
microscope  —  such  as  that  which  is  used  as  a  terrestrial  ocular  in 
the  telescope  —  but  is  very  exceptional.  In  practically  every 
other  converging  -system  the  negative  principal  points  will  be 
outside  the  system  and  on  opposite  sides.  Find  these  and  the  two 
principal  focal  points ;  then,  since  flat  wave-surfaces  have  the 
same  curvature  at  the  second  principal  point  as  they  do  at  the 
second  negative  principal  point  except  with  an  opposite  algebraic 
sign,  as  appears  at  once  by  making  Cp  and  C~p  equal  to  zero  in 
(dl)  and  (Wu),  it  follows  that  a  principal  focal  point  is  exactly  half- 
way between  the  principal  point  and  the  corresponding  negative 
principal  point. 

The  method  of  finding  the  four  nodal  points  is  precisely  similar, 
except  that  the  magnifications  are  taken  as  p0  and  —  pQ  instead 
of  1  and  —  1,  as  in  the  case  of  the  four  principal  points. 


APPENDIX  A  203 


VI.       MAGNIFICATION    OP    OPTICAL   SYSTEMS 

The  general  expression  for  the  magnification  is  equation  (e), 
which,  of  course,  can  be  modified  so  as  to  be  given  in  terms  of 
the  distance  of  the  object  from  x0,  or  from  the  first  vertex  of  the 
system,  or  indeed  from  any  other  point  fixed  with  respect  to  the 
system.  But  it  is  not  easy  to  see  that  this  would  be  of  any  prac- 
tical interest.  A  photographer  may  desire  a  definite  ratio  of  the 
image  in  his  camera  to  the  size  of  the  object,  and  no  doubt  he 
could  tell  at  once  how  far  the  object  must  be  from  the  camera  to 
give  this  ratio,  if  he  had  determined  the  value  of  k  for  two  de- 
terminate positions  of  XQ  and  x^ ;  but  in  practice  his  method  of 
moving  the  instrument  with  respect  to  the  object  until  the  image 
becomes  of  the  desired  size  would  involve  no  more  measurements 
than  that  of  a  single  distance,  which  would  be  also  necessary  in 
the  more  recondite  method. 

For  instruments  used  as  aids  to  vision,  however,  the  expression 
for  magnification  becomes  particularly  interesting,  or  rather  the 
expression  for  angular  magnification,  since  we  care  nothing  for 
the  absolute  size  of  any  image  in  question. 


d ^ 


D ^ 

FIGURE  50. 

Let  Figure  50  represent  any  optical  apparatus  to  be  used  as  an 
aid  in  seeing  o'.  Let  a°  be  an  object  in  such  a  place  that  its  image 
is  at  a^  and  very  near  the  place  of  the  eye.  Call  the  distance 
from  the  eye  to  the  object  Z>,  and  the  distance  from  a°  to  the  ob- 
ject d,  as  represented  in  the  figure.  If  d  is  very  large,  the  instru- 
ment is  called  a  telescope ;  if  small,  a  microscope ;  when  neither 
one  nor  the  other,  we  have  no  name  for  it.  For  example,  the  in- 
struments employed  as  optical  aids  in  reading  the  distant  circles 
of  an  equatorial  may  be  called  with  equal  propriety  telescopes  or 


204  LIGHT 

microscopes.  In  short,  there  is  no  precise  distinction  between 
the  two  types,  and  a  general  equation  of  the  magnifying  power 
ought  to  be  applicable  as  well  to  one  as  the  other. 

We  will  define  the  magnifying  power  of  such  an  instrument  as 
the  ratio  of  the  apparent  dimensions  of  a  small  object  in  the  axis, 
as  seen  through  the  instrument,  to  that  of  the  same  object  seen 
without  the  instrument. 

From  the  diagram  we  see  that  the  angular  dimension  of  o\  as 

measured  from  a°  is  -  ,  which  is  also  twice  the  angle  <£0  for  the 
d 

point  of  the  object  furthest  from  the  axis.  The  angular  subtense 
of  the  object,  as  seen  from  a\  through  the  instrument  is  2^ 

while  the  value  of  this  angle  without  the  instrument  is  —  ,  as  is 

evident  from  the  figure.  This  angle  we  call  2$>.  The  magni- 
fying power  is  therefore  equal  to 


But  from  (g)  we  see  that  </>!  =  p0  ^  °-=  ;  whence,  dividing  by  the 

ou  i  d 

value  of  <£>,  the  magnifying  power  becomes 

a°  D 

PO  "ft-  T 

a°!  d 
as  a  perfectly  general  expression. 

For  the  telescope  proper,  p0  =  1  and  _  =  1     also,   since     the 

d 

length  of  the  instrument  is  negligible  with  respect  to  the  distance 

of  the  object.     In  this  case  the  expression  reduces  to—  ,  or,  if 

a  ! 

we  make  a°  equal  to  the  diameter  of  the  objective,  it  is  equal  to 
the  quotient  derived  by  dividing  the  diameter  of  the  objective 
by  the  diameter  of  its  image  produced  by  the  ocular,  an  old  rule 
first  given,  I  believe,  by  Ramsden.1 

For  the  microscope  the  magnifying  power  is  defined  somewhat 
differently  from  our  foregoing  definition,  namely,  as  the  ratio  of  the 
apparent  size  to  the  size  as  seen  at  a  distance  of  ten  inches,  which 

10ln- 
is  —  _  times  as  great.     Thus  modified,  the  expression  becomes 

1  This  will  be  recognized  as  the  mathematical  proof  of  the  principle  de- 
veloped on  page  115. 


APPENDIX  A 


205 


since  p°  becomes  the  same  as  the  index  of  refraction  of  the  "  im- 

mersion fluid  "  employed.     Not  only  this,  but  we  have  also  - 

2  '/ 

equal  to   the  tangent  of  one-half  the   so-called  "angular  aper- 
ture," whence  the  true  aperture  equals 


~J  a°  > 

25}- 


It  has  long  been  known  that  the  complete  optical  power  of  a  tele- 
scope —  that  is,  both  its  magnifying  power  and  resolving  power  — 
could  be  determined  by  two  linear  measurements,  the  a°  and  a\  of 
our  discussion,  but  it  has,  perhaps,  not  been  suspected  before  that 
with  three  linear  measurements  it  is  possible  to  determine  both 
the  magnifying  power  and  resolving  power  of  a  microscope. 

To  illustrate  the  application  of  these  formulas,  we  may  quote 
the  determinations  of  the  optical  constants  of  a  Zeiss  microscope 
employed  with  ocular  No.  2,  shortest  tube,  and  three  different 
objectives.  The  objectives  were,  a  Zeiss  A,  of  which  his  cat- 
alogue gives  magnification  52  and  numerical  aperture  0.20,  a 
£in-  objective  made  by  me,  and  a  ^irt-  Wales  water-immersion. 
The  measures  were  made  by  placing  a  glass  scale  upon  the  table 
of  the  microscope  and  bringing  the  objective  into  contact  with 
it ;  the  number  of  divisions  of  the  scale  visible  above  the  ocular, 
and  also  the  absolute  length  of  its  image  were  then  recorded,  the 
former  length  being  a°  and  the  latter  a°r  Then  the  tube  was 
raised  a  measured  distance  (d)  until  the  scale  was  in  focus.  In 
the  table  the  measures  are  in  millimetres.  M  is  the  calculated 
magnification,  and  A  the  aperture. 


Objective. 

n. 

a?. 

oV 

d. 

M. 

A, 

Zeiss  A  

1.00 

3.5 

2.0 

85 

52 

.201 

iin-                             .     .     . 

1  00 

3.3 

1.7 

3.5 

140 

.435 

•j^1"'  immersion    .... 

1.33 

1.1 

1.5 

0.48 

351 

0.99 

It  should  be  noted  that,  in  order  to  determine  M  with  preci- 
sion, the  ratio  of  a°  to  a°i,  near  the  axis,  should  be  taken,  since  the 


206  LIGHT 

surface  a°  of  Figure  50  is  not  plane,  but  a  portion  of  a  wave- 
surface.  For  small  apertures  this  distinction  is  insignificant. 

To  investigate  the  resolving  power  of  an  optical  apparatus  used 
in  conjunction  with  the  eye,  we  may  proceed  as  follows  :  — 

Suppose  that  we  have  a  plane  area  at  a  distance  of  ten  inches 
from  the  eye  divided  into  regular  spaces,  and  that  we  observe  it 
through  a  hole  of  diameter  a^.  The  smallest  angular  values  of 
the  elements  of  which  the  field  is  made  and  which  can  still  be 

-jin. 

distinguished  as  separate  elements  is  about  4".5  _   if  a\  is  meas- 

es0! 

ured  in  inches.  This  is  proved  in  all  works  on  the  wave  theory 
of  light  and  is  in  accordance  with  experience.  Now  this  sup- 
position corresponds  precisely  with  the  conditions  of  vision 
through  an  optically  perfect  apparatus  of  the  type  under  con- 

a° 

sideration,  save  that  the  field  is  magnified  in  the  ratio  —  r  for  the 

a  i 

a°  10in- 

telescope  and  n  _  --  in  those  systems  where  d  is  not  indefinitely 
a  i    cL 

great,  consequently  in  such  cases  the  fineness  of  division  of  the 
field  may  be  increased  in  these  ratios.  Hence  the  defining  power 
of  a  telescope  may  be  expressed  by 


and  in  the  other  class  by 


-4"P      d 


10  n  a° 


The  former  of  these  equations  is  a  familiar  one,  and  need  not 
be  further  discussed,  but  the  latter  contains  the  whole  theory  of 
the  defining  power  of  the  microscope,  and  is  therefore  worthy  of 
a  brief  consideration. 

From  what  appears  in  the  discussion  of  the  relation  of  a\  to 
a°,  we  see  that  the  former  is  the  image  of  the  latter,  and  also  that 
«.°,  being  a  portion  of  the  incident  wave-surface,  is  not  plane  as 
is,  very  nearly,  a\.  We  see,  moreover,  that  the  greatest  possible 
diameter  of  a°  is  2d,  in  which  case  the  incident  wave-surface 
would  be  hemispherical  ;  whence  the  maximum  possible  resolving 
power  of  a  microscope  is 

4".5 


APPENDIX  A  207 

To  reduce  this  to  linear  value  we  have  only  to  multiply  by  10in- 
whence  we  have,  as  the  closest  lines  which  can  be  resolved  by  a 
microscope, 

2".25  ^'=0ln.  000011  -; 
n  n 

in  other  words,  the  finest  divisions  which  can  be  seen  with  any 
microscope  in  which  the  objective  is  "dry"  are  about  100,000  to 
the  inch,  while  for  a  "  homogeneous  immersion "  objective  this 
number  may  rise  to  150,000.  Since  the  greatest  value  known  for 
n  in  any  transparent  medium  is  about  2.5,  we  may  say  with  cer- 
tainty that  there  is  no  hope  of  ever  making  a  greater  number  of 
lines  than  a  quarter  of  a  million  to  the  inch  visible. 


VII.     DERIVATIVE    EQUATIONS 

The  most  important  of  the  derivative  equations  in  the  design- 
ing of  optical  apparatus  are  those  which  depend  upon  the  color  or 
variation  of  refrangibility  of  light,  since  they  define  immediately 
the  conditions  for  achromatism.  They  are  of  such  simplicity  in 
our  system,  even  when  rigidly  accurate,  that  it  is  doubtless  worth 
while  to  give  them  for  the  sake  of  completeness,  although  it  is 
outside  the  aim  of  this  sketch  of  geometrical  optics  to  develop 
them  at  length  or  to  do  anything  more  than  indicate  their  use. 

We  will  suppose  the  values  of  the  indices  of  refraction  for  all 
the  media  which  form  the  lens  system  to  be  known  for  a  suffi- 
cient number  of  different  wavelengths  ;  then  it  will  be  possible 
to  express  each  as  a  function  of  some  one  index  (n)  chosen  as  the 
arbitrary  variable.  With  this  as  a  starting-point  it  is  easy  to 
prove  that,  in  general, 


whence  we  may  write  out  immediately  the  derivative  equations 
from  the  system  (a)  as  follows  :  — 

p   ,    /  No    dcf 


208  LIGHT 


It  is  obvious  that  the  condition  for  ordinary  achromatism  is 
that  the  last  equation  should  reduce  to  zero.  We  may  leave  the 
subject  with  the  remark  that  the  second  derivatives,  useful  in  the 
consideration  of  secondary  chromatic  aberrations,  are  hardly  more 
complex  than  these. 


APPENDIX  B 

THE  following  considerations  may  add  something  to  our 
knowledge  of  this  interesting  phenomenon  of  scintillation.  It 
is  quite  evident  that  with  altitude  above  the  earth's  surface 
the  atmosphere  is  neither  optically  homogeneous  nor  regularly 
varying  in  refractive  power;  in  short,  it  is  always  more  or  less 
irregular.  Imagine  a  point-source  of  light,  like  an  enormously 
brilliant  star,  outside  the  atmosphere.  This  might  be  expected  to 
illuminate  the  surface  of  the  earth  in  a  highly  irregular  manner 
on  account  of  the  infinite  optical  irregularity  of  the  air.  The 
illumination  might  be  conceived  as  distributed  in  a  mottled  or 
reticulated  manner,  something  like  the  distribution  of  light  on 
a  wall  upon  which  a  distant  electric  light  is  shining  through  a 
window  of  ordinary  glass,  although  any  valid  estimate  regarding 
the  size  and  density  of  such  a  reticulation  would  seem  quite 
hopeless.  There  is  no  such  source  of  light;  even  the  planet 
Venus  at  its  brightest  is  not  sufficiently  bright  to  cast  obvious 
shadows.  But  in  rare  instances  we  are  able  to  observe  the  result 
of  the  illumination  by  a  very  bright  linear  source  of  light, 
namely,  in  a  total  solar  eclipse  immediately  before  the  second 
and  after  the  third  contacts.  It  is  quite  evident  that  in  this 
case  the  suppositions  shadows  would  preserve  their  character 
only  in  a  direction  at  right  angles  to  the  direction  of  elongation 
of  the  source.  Moreover,  it  is  equally  evident  that  all  apparent 
motion  of  these  shadows  would  be  at  right  angles  to  their 
lengths,  since  a  purely  linear  object  does  not  betray  a  dis- 
placement in  the  direction  of  its  length.  Consequently,  such 
shadows,  if  existent,  ought  to  appear  as  more  or  less  distinct 
and  parallel  bands  drifting  with  the  layers  of  the  atmosphere  in 
which  they  have  their  origin,  and  hence  with  the  most  diverse 
velocities;  but  apparently  always  moving  perpendicularly  to 
their  lengths.  The  intervals  which  separate  them  may  well 

U 


210  LIGHT 

have  almost  any  value  ;  but  only  a  limited  range  of  values, 
perhaps  from  a  few  inches  to  a  moderate  number  of  feet,  would 
be  likely  to  attract  attention.  This  is  an  accurate  description  of 
the  famous  "  shadow  bands  "  attending  total  eclipses,  at  least 
of  those  observed  by  me  on  several  occasions. 

We  are  not  limited  to  this  particular  phenomenon  for  proof  of 
the  general  correctness  of  the  view  given  in  the  preceding  para- 
graph. If  one  could  observe  for  a  very  brief  time,  say  during 
a  single  thousandth  of  a  second,  the  minute  appearance  of  the 
sun,  he  might  expect  to  detect  variations,  either  in  relative 
brightness  or  in  sharpness  of  definition,  in  the  appearance  of  the 
surface,  which  ought  to  be  arranged,  generally  speaking,  in  a 
roughly  reticulated  way.  But  such  an  observation  would  be 
hopeless  if  restricted  to  the  unaided  eye,  for,  although  there 
would  be  no  difficulty  in  securing  a  perfectly  satisfactory  visual 
impression  in  the  time  designated,  any  ill-defined  detail  upon 
such  a  limited  area  as  is  occupied  by  the  sun  would  surely 
escape  detection.  On  the  other  hand,  the  use  of  a  telescope 
greatly  modifies  the  problem  because  then,  instead  of  a  projec- 
tion of  certain  atmospheric  irregularities  from  the  very  small 
area  of  the  pupil  of  the  eye  which  may  be  considered  as  prac- 
tically a  point,  one  observes  that  from  the  considerable  area  of 
the  objective.  Nevertheless,  for  those  atmospheric  irregularities 
which  are  remote  from  the  observer,  say  from  a  mile  or  two  to  a 
hundred  miles,  the  telescope  also  may  be  regarded  as  of  insignifi- 
cant diameter,  and  hence  the  foregoing  reasoning  may  be  extended 
from  the  case  of  the  unaided  eye  to  this  case.  In  some  remark- 
able photographs  of  the  sun  taken  by  Janssen  a  number  of  years 
ago  we  find  observational  proof  of  these  inferences,  of  a  most 
convincing  character.  With  these  photographs  the  exposure 
was  sufficiently  brief  for  the  purpose,  and  the  optical  power  was 
great  enough  to  exhibit  the  fine  granular  structure  of  the  photo- 
sphere. They  show  just  such  a  reticulated  variation  in  definition 
of  details  as  would  fit  the  description  here  given,  and,  as  they 
were  in  no  two  cases  alike,  astronomers  had  no  hesitation  in 
referring  the  irregularities  to  our  atmosphere  rather  than  to  a 
solar  origin  which  Janssen  favored. 

If  the  general  validity  of  the  foregoing  assumptions  is 
admitted,  the  familiar  phenomena  of  scintillation  are  easily 
accounted  for.  In  the  case  of  great  and  local  irregularity  of  the 


APPENDIX  B  211 

atmosphere  we  may  be  sure  that  even  relatively  small  volumes 
of  air  would  include  variations  of  density,  and  that,  owing  to 
continuous  changes,  the  stars  would  appear  to  the  unassisted 
eye  to  undergo  rapid  and  extreme  alterations  of  brightness.  In 
a  large  telescope  the  image  of  a  star  at  any  instant  would  re- 
semble that  formed  by  a  telescope  of  which  the  material  of  the 
objective  is  defective;  but  this  would  change  with  great  rapidity 
and  thus  appear  to  vary  largely  in  size  and  shape  though  little 
in  brightness.  The  changes  would  be  chiefly  due  to  bodily 
motion  of  the  air,  either  as  wind  or  convection  currents.  Chro- 
matic changes  in  stellar  images  would  be  relatively  insignificant, 
since  the  paths  of  different  wavelengths  of  light  would  not  be 
widely  distributed  in  the  neighborhood  of  the  observer.  During 
exceptionally  quiescent  states  of  the  atmosphere  these  results  of 
irregular  refractions  would  be  considerably  modified.  On  such 
occasions  the  effective  irregularities  would  be  not  only  less 
pronounced,  but  also  generally  much  further  from  the  observer. 
Two  consequences  to  be  deduced  from  the  remoteness  of  the 
source  of  disturbances  are  evident:  First,  the  distinction  be- 
tween the  effect  as  observed  by  the  naked  eye  and  by  a  telescope 
must  largely  disappear;  and,  second,  chromatic  phenomena 
must  become  more  conspicuous,  since  an  atmospheric  irregu- 
larity which  may  greatly  decrease  the  amount  of  red  light  that 
comes  to  the  observer  at  a  given  instant  lies  quite  remote  from 
the  path  of  the  blue  light  from  the  same  source.  A  third  con- 
sequence is  important  even  if  less  obvious.  The  motion  of  such 
a  disturbing  body  of  air  relative  to  the  stars  will  be  chiefly  due 
to  the  rotation  of  the  earth  rather  than  to  the  wind,  because 
the  apparent  motion  of  the  stars  referred  to  any  portion  of  the 
earth  is  independent  of  the  distance  of  the  point  of  reference, 
while  the  apparent  motion  of  the  wind  decreases  directly  as  its 
distance  from  the  observer.  The  fact  that  such  regular  scintil- 
lation is  observed  only  during  moderately  calm  weather  is  a 
further  reason  for  this  last  conclusion.  The  explanation  of  the 
remarkable  observations  of  B-espighi  on  the  variations  in  the 
spectra  of  scintillating  stars,  briefly  described  in  the  text,  is 
deducible  at  once  from  these  assumptions  and  has  already  been 
given. 


APPENDIX  C 

IN  the  seventh  chapter  of  this  book  the  description  and  theory 
of  halos  has  been  given  with  considerable  extension,  because 
this  class  of  phenomena  presents  us  with  many  interesting  prob- 
lems, including  a  number  still  unsolved,  which  have  engaged  the 
attention  of  philosophers  for  centuries.  The  most  extensive 
work  on  the  subject  of  halos  which  has  appeared  is  a  memoir 
by  Professor  Bravais,  contained  in  the  Journal  de  Vficole  Royale 
Poly  technique,  tome  xviii.,  1847.  This  author  performs  a  cap- 
ital service  in  collecting  the  available  data  of  a  vast  number 
of  widely  scattered  observations  extending  over  a  period  of 
more  than  two  centuries;  but  in  the  second  of  his  aims  to  find 
a  consistent  physical  explanation  of  all  their  features,  he  is  far 
from  being  successful.  Notwithstanding  the  somewhat  obvious 
faults  in  the  general  theory  here  developed  and  applied,  Bra- 
vais's  work  has  been  almost  universally  regarded  as  complete 
and  practically  final,  so  that  for  more  than  half  a  century  abso- 
lutely nothing  has  been  added  or  altered.  We  can  go  even 
further  than  the  statement  made  concerning  the  theoretical 
explanations  in  this  widely  quoted  work,  and  assert  that  not  a 
single  explanation  originating  with  its  author  will  resist  criti- 
cal analysis.  Bravais's  fundamental  assumptions,  which  he 
shares  in  common  with  all  his  immediate  predecessors  in  this 
field  —  namely,  that  elongated  prisms  will  fall  through  the  air 
in  a  vertical  position  and  that  flat  ones  will  fall  edgewise  —  are 
mechanically  unsound.  Thus  only  those  features  of  halos  which 
are  independent  of  the  ratio  of  length  to  breadth  of  the  prisms 
can  be  possibly  explained.  But  at  the  time  that  Bravais  under- 
took his  work  everything  which  could  be  explained  on  the 
wholly  correct  assumptions  that  we  have  in  fact,  besides  hex- 
agonal crystals  of  ice  with  fortuitously  directed  axes,  two  other 
groups  with  horizontal  and  vertical  axes,  respectively,  had  been 


214  LIGHT 

completely  developed.  As  long  as  the  unsoundness  of  the  addi- 
tional hypotheses  remained  unquestioned  there  was  no  way  open 
to  account  for  the  residual  phenomena,  except  the  assumption 
of  the  action  of  more  complex  crystalline  forms.  In  this  Bravais 
showed  a  boldness  approaching  temerity,  for  he  apparently  telt 
justified  in  resting  an  explanation  upon  any  form  not  absolutely 
excluded  by  the  laws  of  crystallography.  Thus  he  found  him- 
self obliged  to  assume  the  presence  of  about  a  score  of  different 
forms,  some  being  elongated  prisms  with  intricate  stellar  cross- 
sections,  others  with  varied  pyramidal  termini,  and  even  rhom- 
bohedra.  None  of  these  has  ever  been  observed.  If  any  fact 
concerning  halos  can  be  predicated  a  priori,  it  is  that  they  must 
be  due  to  very  simple  crystals,  since  the  number  present  is  not 
only  enormously  great,  but  because  any  given  feature  would  be 
weakened  and  perhaps  totally  suppressed  by  the  presence  of  a 
large  number  of  bodies  suspended  in  the  atmosphere,  not 
engaged  in  its  production.  In  the  discussion  of  these  phenom- 
ena we  have  assumed  the  existence  of  two  types  of  ice  crystals 
only,  and  both  of  these  have  been  observed  innumerable  times. 
Moreover,  they  are  the  simplest  crystal  forms  known  for  this 
material.  In  the  following  pages  we  purpose  to  complete  and 
summarize  the  explanations  given  in  this  book. 

Of  all  the  extraordinary  halos  recorded  the  most  perfectly 
developed  and  most  complicated  one  is  that  described  by  Lowitz 
in  the  Nova  Acta  Academice  Petrop.,  tomus  viii. ,  which  he  observed 
at  St.  Petersburg  on  June  29,  1790.  All  three  groups  of 
directed  crystals  were  present  and  effective  on  this  occasion. 
Unfortunately  his  drawing  has  the  defect  of  possessing  no  recog- 
nizable system  of  projection,  and,  what  is  still  more  to  be  re- 
gretted, there  is  no  certain  indication  of  the  changes  dependent 
on  the  sun's  varying  altitude,  everything  seen  from  half-past 
seven  in  the  morning  until  noon  being  represented  in  the  one 
diagram.  Notwithstanding  these  shortcomings,  so  important 
to  the  investigator  of  a  physical  theory,  they  should  not  blind 
us  to  the  fact  that  the  observation's  are  of  remarkable  merit  and 
may  well  excite  our  astonishment  that  so  much  was  seen  and 
accurately  recorded  with  entire  lack  of  guidance  by  theory. 
Bravais's  copy  of  this  figure  is  far  from  satisfactory,  and,  as 
the  original  is  not  very  accessible,  it  is  quite  worth  while  to 
give  a  faithful  reproduction  here,  particularly  since  it  may  be 


APPENDIX  C 


215 


regarded  as  containing  all  the  remaining  features  of  the  halo, 
which  are  certainly  established  by  authentic  observations  and 
not  included  in  the  halo  of  Parry  and  Sabine. 

The  description  accompanying  this  figure  runs  as  follows :  — 
"1.    The  sun  was  surrounded  by  two  circles  ebdk  and  eide, 


FIGURE  51. 

which  intersected  each  other  at  d  and  e,  and  of  which  the  color 
was  red  on  the  side  toward  the  sun  and  whitish  on  the  opposite 
side.  The  sun  was  at  a  between  their  centres  a  and  (3.  The 
exterior  arcs  dbe  and  dee  were  much  more  distinct  and  brilliant 
than  the  interior  dke  and  die.  The  resemblance  of  the  colors 
ga,ve  to  these  last  an  elongated  figure  or  oval,  and  the  two 
others  had  the  form  of  a  circle  flattened  above  and  below.  At 
the  upper  point  of  their  intersection  a  considerable  part  wdw 


216  LIGHT 

was  observed  of  an  extraordinary  vividness,  of  which  the 
splendor  was  almost  as  dazzling  to  the  eye  as  that  of  the  sun 
itself. 

"2.  The  inferior  point  of  intersection  e  touched  an  inverted 
semicircular  arc  ref  very  bright  and  wide,  which  was  in  relation 
to  its  diameter  the  smallest  of  all  the  arcs. 

"3.  Another  circle  zzz,  similarly  colored  but  at  a  greater 
distance  from  the  sun  and  imperfect  toward  the  horizon,  which 
an  arc  above  pzq  gave  the  appearance  of  being  horned,  had  the 
sun  itself  for  a  centre. 

"4.  This  last  circle  was  tangent  toward  the  south  and  toward 
the  east  to  two  inverted  arcs  tt  and  vv,  with  respect  to  the  sun, 
exactly  like  portions  of  a  rainbow,  not  only  from  their  size  but 
also  from  the  intensity  of  their  prismatic  colors,  in  which  they 
were  distinguished  from  all  the  other  arcs  of  this  meteor. 
Buildings  prevented  me  from  seeing  whether  they  extended 
quite  to  the  horizon. 

"5.  Still  another  circle  complete  and  white  afhga  which 
enclosed  a  very  large  area,  was  parallel  to  the  horizon,  which 
it  followed  completely  around,  and  which  consequently  had  the 
zenith  for  a  centre.  In  the  circumference  of  this  one  lay  the 
sun  itself  and  five  parhelia,  of  which  three/,  h,  and  g  opposite 
the  sun  toward  the  northwest  were  white  and  faint,  and  the  two 
others  x  and  y  at  each  side  of  the  sun,  in  the  southeast,  colored 
and  very  bright. 

"6.  These  two  last  parhelia  which  were  at  some  distance 
from  the  intersections  of  the  great  horizontal  circle  by  the  two 
coronas  which  surrounded  the  sun,  sent  in  the  first  place  from 
the  two  sides  very  short  colored  arcs  xi  and  yk  of  which  the 
direction  was  inclined  below  the  sun  so  as  to  reach  the  interior 
semicircular  arcs  die  and  dke.  In  the  second  place  they  had 
long  tails,  bright  and  white  x£  and  y^  directed  away  from  the 
sun  and  included  in  the  circumference  of  the  great  circle  afhg. 

"  7.  Finally  two  great  circular  arcs  dlh  and  dmh  of  a  white 
color  appeared  inside  the  great  horizontal  circle  afhg,  but  they 
were  so  faint,  that  several  persons  to  whom  I  attempted  to  show 
them,  were  unable  to  perceive  them.  In  the  first  place  they  met 
near  the  sun  at  d  in  the  dazzling  brightness,  and  they  again 
crossed  each  other  on  the  other  side,  as  well  as  the  great  circle, 
in  the  centre  of  the  pale  parhelion  h,  whence  they  extended 


APPENDIX   C  217 

sensibly  further,  several  degrees  beyond  the  great  circle  toward 
the  northwest  point  of  the  horizon,  to  n  and  0. 

"Such  was  this  beautiful  meteor  at  ten  o'clock  in  the  morning 
when  it  had  attained  its  greatest  perfection.  With  regard  to 
the  successive  changes  of  its  parts,  I  made  the  following  obser- 
vations in  addition : 

"  At  half-past  seven  in  the  morning  the  two  coronas  around  the 
sun  dbek  and  dcei  were  not  yet  perfect,  only  the  interior  arcs  die 
and  dke  appeared  under  the  form  of  a  perfect  oval  with  a  bright 
region  above  at  d.  It  was  only  little  by  little  that  this  bright- 
ness extended  itself  on  both  sides,  in  the  form  of  the  arcs  dw, 
dw.  which  became  larger  and  larger,  until  they  united  finally  at 
nine  o'clock  at  e  near  the  brilliant  semicircle  ref. 

"That  which  was  most  remarkable  in  these  two  circles  or 
coronas,  which  intersected  each  other,  was  the  fact  that  after 
having  attained  their  perfection  they  approached  each  other 
more  and  more  until  in  the  end  they  formed  only  a  single 
corona  which  had  the  sun  at  its  centre.  Nevertheless  the  upper 
and  lower  portions  always  retained  a  very  sensible  brightness. 
About  this  time  the  two  arcs  with  prismatic  colors  tt  and  vv 
disappeared.  The  two  parhelia  x  and  y  on  the  contrary  sepa- 
rated themselves  more  and  more  widely  from  the  sun  and  finally 
disappeared  entirely  at  a  quarter  to  eleven. 

"  The  great  horizontal  circle  afhga  and  its  three  faint  parhelia 
/,  h,  and  g  still  remained  and  did  not  disappear  until  eleven 
thirty-five.  That  which  still  deserved  being  remarked  in  this 
great  circle,  is  that  preserving  always  the  sun  and  its  five  par- 
helia in  its  circumference,  it  remained  constantly  parallel  to 
the  horizon,  and  consequently  retained  the  zenith  as  a  centre; 
this  is  why  the  more  the  sun  approached  the  meridian,  the  more 
it  rose  above  the  horizon,  the  great  circle  of  which  the  extent 
was  at  first  extremely  large,  when  the  sun  was  more  elevated, 
diminished  continuously,  until  at  the  end  its  diameter  became 
almost  as  small  as  that  of  the  two  united  coronas.  The  same 
thing  happened  to  the  two  arcs  dlh  and  dmh  which  were  within 
the  horizontal  circle.  Their  points  of  intersection  were  always 
at  h  and  d. 

"  At  noon  there  remained  nothing  of  this  beautiful  phenomenon 
except  a  simple  corona  very  bright  at  its  upper  and  lower  por- 
tions, which  finally  disappeared  entirely  at  half-past  twelve. 


218  LIGHT 

"  In  general  this  meteor  was  composed  of  twelve  arcs,  of  which 
nine  were  colored,  in  such  a  way  that  in  all  cases  the  edge 
toward  the  sun  was  red  and  the  opposite  edge  white.77 

There  is  comparatively  little  new  in  this  remarkable  halo, 
that  is,  little  that  has  not  been  already"  described  and  ex- 
plained. Section  1  is  somewhat  obscure  in  ascribing  the  form 
of  the  inner  curve  to  the  resemblance  of  the  colors,  but  it 
becomes  quite  clear  if  we  interpret  it  as  meaning  that  the  two 
arcs  named  seemed  from  their  similarity  to  belong  together. 
This  was  indeed  the  case  because  they  were  halves  of  the  ordi- 
nary twenty-two-degree  circle.  There  is  nothing  surprising  in 
the  fact  that  this  appeared  to  the  observer  to  be  distinctly  oval, 
for  in  a  great  number  of  similar  cases  the  same  error  is  found 
in  the  records.  Even  the  present  writer,  although  quite  aware 
of  the  deception,  found  a  well-developed  circumscribing  oval, 
seen  at  New  Haven  during  the  early  afternoon  of  April  13,  1901, 
sensibly  round,  with  the  enclosed  twenty-two-degree  circle  — 
here  also  the  less  brilliant  of  the  two  —  appearing  as  an  ellipse. 
In  passing,  it  may  be  noted  that  this  observation  suggests  the 
possibility  that  many  of  the  high-sun  halos,  in  which  the  upper 
and  lower  portions  are  decidedly  brighter  than  the  rest,  are 
really  the  circumscribing  oval  unaccompanied  by  the  ordinary 
twenty-two-degree  circle.  This  is  a  point  which  may  be  worthy 
of  attention  in  the  future. 

Section  2  describes  a  rare  feature,  although  recorded  by 
several  other  observers.  It  is,  without  doubt,  the  inferior 
complement  of  the  arc  xev  of  Parry  and  Sabine,  the  explanation 
of  which  we  have  found  in  the  action  of  the  crystals  of  the  B 
group.  Its  most  characteristic  peculiarity  was  the  smallness  of 
its  apparent  diameter;  but  its  brightness  pretty  certainly  indi- 
cates fixity  of  direction  of  its  faces.  The  record  is  hardly  pre- 
cise enough  to  admit  of  quite  satisfactory  proof  by  calculation, 
nevertheless  it  is  worth  stating  that  an  approximate  calculation 
for  an  altitude  of  the  sun  of  30°,  corresponding  to  about  the 
beginning  of  Lowitz's  observations,  gave  me  a  radius  of  about 
6°  for  the  summit  of  an  arc  more  nearly  resembling  a  par- 
abola than  a  circle.  The  angular  distance  of  the  apex  from 
the  sun  was  found  to  be  24°. 7.  When  we  take  into  account 
the  inevitable  excess  in  eye  estimates  of  angular  magnitudes 


APPENDIX   C  219 

of  objects  near  the  horizon,  this  is  not  at  all  a  bad  agreement 
with  the  drawing,  although  the  radius  of  the  curve  is  there 
approximately  twice  this  value.  A  very  moderate  oscillation 
of  these  crystals  in  their  descent  would  materially  increase 
the  radius  of  the  arc. 

Section  3  describes  the  well-known  forty-six-degree  circle  with 
its  upper  tangent  arc.  In  this  halo  the  evident  paucity  of 
undirected  crystals,  as  evinced  by  the  faintness  of  the  twenty- 
two-degree  halo,  suggests  a  question  concerning  the  origin  of 
the  greater  circle.  It  is  true  that  fortuitous  crystals  should 
produce  such  a  circle,  but  whether  with  sufficient  intensity 
to  make  it  visible  against  a  bright  sky  is  quite  another  ques- 
tion. I  have  never  found  a  trace  of  the  forty-six-degree  circle, 
although  I  have  observed  scores  of  simple  halos,  not  a  few 
of  which  were  remarkably  bright.  On  the  whole,  in  view  of 
the  facts  (a)  that  there  are  six  points  in  the  circumference  at 
which  one  or  another  of  the  directed  crystals  may  contribute 
largely  to  the  light  from  it,  and  (b)  that  a  slight  rocking 
motion  of  the  crystals  at  these  points  would  produce  extended 
arcs  coincident  with  the  circle,  I  am  inclined  to  replace  the  ac- 
cepted explanation  by  referring  the  phenomenon  to  concurrent 
action  of  all  three  groups  of  directed  crystals,  eliminating  the 
fortuitous  crystals  as  wholly  negligible  agents. 

Section  4  gives  us  nothing  new,  since  these  arcs  are  the  same 
as  those  seen  by  Parry  and  Sabine  at  a  slightly  lower  altitude 
of  the  sun.  They  appear  to  have  vanished  about  nine  o'clock, 
at  which  time  the  sun  stood  40°  above  the  horizon. 

Section  5  is  a  description  of  the  parhelic  circle  of  which  the 
explanation  has  been  given  quite  fully  in  the  text. 

Section  6  presents  nothing  new  except  the  singular  oblique 
arcs  extending  from  the  parhelia  to  the  twenty -two-degree 
circle.  No  one  else  has  described  them  with  the  same  particu- 
larity, hence  they  are  very  properly  known  as  the  "  oblique  arcs 
of  Lowitz."  Galle  has  explained  them  satisfactorily  by  referring 
them  to  the  action  of  crystals  similar  to  those  producing  the 
neighboring  parhelia,  but  possessing  a  rocking  or  balancing 
motion  in  their  fall. 

Section  7  describes  the  singular  oblique  arcs  which  have 
proved  such  a  puzzle  to  all  writers  on  this  subject.  According 
to  our  views  they  are  the  visible  portion  of  a  continuous  curve 


220 


LIGHT 


which  owes  its  origin  to  two  refractions  separated  by  an  internal 
reflection  by  crystals  of  class  B.  A  special  solution  for  the 
case  of  the  sun  at  an  altitude  above  the  horizon  of  65°  has  been 
shown  in  Figure  39,  page  152.  The  values  derived  by  calcula- 
tion for  this  particular  halo  are  given  in  the  table  immediately 
following:  — 


A  60° 

90° 

110° 

130° 

133° 

140° 

150° 

160° 

S'27.6 
31.4 

46.1 
27.7 

62.0 
24.1 

83.3 
19.8 

86.2 
18.9 

95.4 
17.2 

110.5 
14.7 

129.8 
126 

S"  92.4 
31.4 

136.1 
27.7 

158.0 
24.1 

176.7 

19.8 

179.8 
18.9 

184.6 
17.2 

189.5 
14.7 

190.2 
12.6 

£"'92.4 
43.1 

136.1 
37.6 

158.0 
32.3 

176.7 
26.4 

179.8 
25.1 

184.6 
22.8 

189.5 
19.4 

190.2 
16.7 

Here  the  figures  in  the  first  horizontal  row  are  the  differences 
between  the  azmuth  of  the  sun  and  the  vertical  plane  containing 
the  poles  of  the  refracting  surfaces;  consequently,  also  the  angle 
between  the  vertical  containing  the  sun  and  the  vertical  base  of 
the  prism  upon  which  the  internal  reflection  takes  place.  The 
rows  attaching  to  S'  give  the  azmuth  and  zenith  distance  of  the 
image  of  the  sun  formed  by  the  first  refraction,  the  upper  figure 
being  the  azmuth  angle.  The  next  two  rows  give  the  co- 
ordinates of  the  images  formed  by  the  subsequent  reflection,  and 
the  rows  attaching  to  S'"  are  the  corresponding  co-ordinates  for 
the  images  formed  by  the  completed  action  of  the  prisms,  and 
are  the  points  through  which  the  double  spiral  of  Figure  39  is 
drawn. 

It  is  very  difficult  to  apply  the  theory  with  any  degree  of  con- 
fidence to  these  observations  of  Lowitz  because  of  their  indefi- 
niteness.  For  example,  the  observer  remarks  that  his  description 
applies  to  the  state  at  ten  o'clock  in  the  morning,  at  which  hour 
the  zenith  distance  of  the  sun  was  only  43°.  This  would  have 
brought  the  zenith  well  within  the  outer  concentric  halo,  which 
ill  accords  with  the  drawing.  At  nine  o'clock  the  sun's  zenith 
distance  was  50°,  which  still  seems  too  high  for  the  drawing. 
It  is  evident  that  we  have  a  wide  range  of  choice  for  the  ele- 
ments of  our  problem,  with  little  hope  of  reproducing  the  com- 
posite drawing  in  the  calculations  founded  upon  these  elements; 


APPENDIX   C  221 

but  if  we  adopt  45°  as  the  zenith  distance  of  the  sun,  the  result 
of  calculations  founded  on  the  theory  presented  in  this  book  is 
as  follows :  The  complete  curve  consists  of  two  spirals  like  those 
of  Figure  39,  which  together  form  two  loops,  one  including  the 
other,  with  the  node  at  the  anthelion.  In  the  present  case, 
however,  the  inner  loop  is  far  larger,  embracing  the  zenith  and 
tangent  to  the  twenty-two-degree  circle  at  its  highest  point.  The 
point  of  tangency  is  not  a  cusp,  as  represented  in  the  drawing,  but 
the  lowest  point  of  a  nearly  circular  arc  of  about  twenty-nine- 
degree  radius  —  a  difference  which  counts  for  nothing,  since  this 
portion  of  the  curve  would  be  quite  invisible  on  account  of  the 
overwhelming  brightness  of  the  arcs  wdw.  The  two  branches 
of  the  curve  intersect  at  an  angle  of  very  nearly  60°,  which 
agrees  with  the  drawing.  From  the  anthelion  the  two  branches 
extend  with  continuously  diminishing  brightness  and  curvature 
until  they  meet  at  a  point  34°  below  the  sun.  The  inner  loop 
of  the  calculated  curve  embraces  somewhat  more  area  with 
respect  to  that  surrounded  by  the  parhelic  circle,  but  this  is 
chiefly  due  to  the  difference  in  shape  in  the  region  between  the 
zenith  and  the  sun.  It  is  a  singular  result  of  the  calculations 
that,  whereas  the  spirals  of  Figure  39  were  produced  by  light 
incident  on  one  of  the  upper  oblique  faces  of  the  B  crystals, 
in  this  case  only  the  outer  loop  is  thus  formed,  the  inner  one 
being  produced  by  an  entering  refraction  on  the  upper  horizontal 
surface  and  a  final  emergence  from  a  lower  oblique  face. 

As  this  completes  the  explanation  of  all  known  features  of 
the  complex  phenomenon  called  the  halo,  it  may  be  well  to  col- 
lect them  in  tabular  form.  We  will  first  give  those  of  which 
the  origin  has  been  known  for  a  longer  or  shorter  time,  with 
the  name  of  the  physicist  who  first  found  the  true  explanation. 

1.  Halo  of  twenty-two-degree  radius.     MAKIOTTE. 

2.  Parhelia  of  22°.     MAKIOTTE. 

3.  Oblique  arcs  of  Lowitz.     GALLE. 

4.  Tangent  arcs  to  twenty-two-degree  halo,  which  become  the 
circumscribing  oval  with  high  sun.     YOUNG  and  VENTURI. 

5.  Halo  of  forty-six-degree  radius.     CAVENDISH.     (Unless  ob- 
jections given  on  page  219  in  regard  to  this  feature  are  valid.) 

6.  Horizontal  tangent  arcs  to  forty-six-degree  halo.     GALLE  j 
perfected  by  BRAVAIS. 


222  LIGHT 

To  these  must  be  added  the  following  which  have  not  hitherto 
been  explained  at  all,  or  wrongly  explained  because  grounded 
upon  theories  which  are  untenable  :  — 

7.  Lateral  tangent  arcs  to  the  forty-six-degree  halo. 

8.  Parhelic  circle. 

9.  Paranthelia. 

10.  Anthelion. 

11.  The  arcs  above  and  below  the  twenty -two-degree  halo. 

12.  The  short  oblique  arcs  through  the  anthelion. 

13.  Spiral  arcs  through  the  anthelion. 

14.  Vertical  columns. 

There  is,  however,  a  celebrated  halo  that  contains  a  feature 
not  mentioned  in  the  list,  which  has  given  a  great  deal  of  trouble 
to  writers  on  this  subject  from  the  time  of  Huyghens  down. 
It  is  a  rather  remarkable  halo  observed  by  Hevelius  in  1661, 
and  described  fully  in  Smith's  Opticks,  vol.  i.,  pp.  221,  222, 
although  with  the  exception  of  this  feature  it  seems  to  have 
been  a  well-developed  halo  depending  upon  the  presence  of  the 
A  group  for  its  chief  characteristics.  The  exceptional  feature 
is  a  circle,  of  which  only  the  lower  portions  are  shown  in  the 
figure  illustrating  it,  everywhere  90°  from  the  sun,  and  there- 
fore a  great  circle.  Bravais,  who  styles  this  as  the  most 
authentic  of  all  extraordinary  halos,  cites  all  the  explanations 
offered,  points  out  their  fallacies,  but  quite  frankly  declares  his 
inability  to  propose  any  more  satisfactory  theory.  Since  I  am 
forced  to  follow  Bravais  exactly  in  this  respect,  it  may  be  well 
to  review  the  evidence  of  the  existence  of  the  ninety-degree 
circle,  beyond  that  contained  in  the  original  record.  There  is 
nothing  in  the  records  of  the  time  since  Bravais  which  bears 
upon  this  point,  at  least  a  search  by  me  has  led  to  no  result  ; 
v  hence  we  are  confined  to  the  three  examples  which  that  author 
finds. 

The  first  is  found  in  the  description  of  the  halo  observed  at 
Melville  Island,  given  by  Parry  and  Sabine.  The  passage 
in  the  last  paragraph  of  the  quotation,  page  143,  describing 
the  faint  light  about  a  quadrant  from  the  sun,  is  taken  as 
an  observation  of  the  circle  in  question;  but  a  most  casual 
reading  demonstrates  that  such  an  interpretation  is  an  entire 
misapprehension. 


APPENDIX   C  223 

The  second  instance  is  found  in  a  very  uncritical  description 
of  a  halo  seen  at  Derby  in  England,  in  1802,  and  published  in  the 
"  Philosophical  Magazine,"  vol.  xii.,p.  373.  In  this  case  neither 
the  name  of  the  observer  nor  the  place  of  the  sun  in  the  heavens 
is  given.  The  passage  in  which  Bravais  finds  evidence  of  the 
ninety-degree  circle  reads  as  follows :  " .  .  .  the  fourth  [circle] 
circumscribed  all  the  others,  and  was  touched  upon  the  western 
side  by  part  of  another  of  the  same  diameter."  It  is  quite  clear 
that  this  circle  did  not  have  a  radius  of  90°,  not  only  because  no 
ordinary  observer  would  dream  of  calling  a  great  circle  of  which 
the  sun  occupies  the  position  of  one  pole,  a  circumscribing  circle, 
but  also  because  in  that  case  another  circle  tangent  to  it  and  of 
the  same  diameter  would  be  identical  with  it.  Unquestionably, 
this  fourth  circle  was  the  forty-six-degree  halo,  and  the  circle 
touching  it  was  the  upper  tangent  arc. 

The  final  case  appears  to  be  much  more  conclusive.  It  is 
that  of  a  lunar  halo  observed  by  Erman  in  Siberia,  in  1828. 1 
Here,  with  the  most  minute  particularity,  that  traveller  gives 
the  results  of  his  observations,  together  with  the  fact  that  at 
10.30  P.  M.,  Tobolsk  mean  time,  the  measured  distance  of  the 
moon  from  the  vertex  of  an  auroral  arch  was  83°. 2;  moreover, 
that  at  the  same  instant  the  lunar  halo  intersected  the  auroral  arch 
a  few  degrees  to  the  west  of  its  vertex.  This  seems  very  con- 
vincing as  to  the  existence  of  a  halo  with  a  radius  of  85°  to  90° ; 
but  reference  to  the  details  of  the  original  account  shows  certain 
peculiarities  which  cannot  fail  to  awaken  strong  doubts  con- 
cerning this  conclusion.  In  the  first  place,  Erman  describes 
the  halo  without  any  intimation  that  it  is  an  unusual  one.  Then 
he  mentions  the  fact  that  it  coincided  with  a  part  of  one  of  a 
system  of  Concentric  arcs  which  are  supposed  to  be  auroral  on 
account  of  their  fixity  of  position  with  respect  to  the  earth. 
Finally,  he  gives  the  measured  distance  of  the  moon  from  the. 
apex  of  the  lowermost  arch  at  6.30  in  the  evening,  which  he 
found  to  be  86°.  At  this  time  the  moon  was  close  to  the 
horizon;  consequently,  if  the  radius  of  the  halo  was  90°,  it 
would  have  intersected  all  the  auroral  arches  nearly  orthago- 
nally,  and  a  partial  coincidence  at  any  point  would  have  been 
quite  out  of  the  question.  But  this  is  not  the  only  incon- 

1  Erman,  Reise  um  die  Erde,  vol.  i.,p.  544. 


224  LIGHT 

sistency.  An  investigation  as  to  the  position  of  the  moon  at  the 
place  given  and  at  the  epoch  of  November  24,  1828,  10.30  p.  M., 
shows  that  its  true  distance  from  the  point  indicated  as  marking 
the  place  of  the  vertex  of  the  auroral  arch  was  107°  j  hence 
Erman's  statement  is  erroneous. 

But  it  is  quite  easy  to  supply  to  the  printed  account  an  emen- 
dation which  eliminates  all  the  difficulties  and  contradictions. 
We  find  that,  on  the  evening  in  question,  the  distinguished 
traveller  was  at  Sawodinsk,  a  place  2°  north  of  Tobolsk, 
engaged  in  making  a  complete  and  protracted  set  of  observa- 
tions on  the  magnetic  elements  of  the  place.  During  the  inter- 
vals of  these  observations  —  important  as  a  part  of  a  very 
elaborate  system  —  he  entered  in  his  notebook  the  contemporary 
phenomena  of  auroral  arches  and  the  halo.  At  the  later  hour 
named  he  made  the  angular  measure,  probably  with  a  sextant. 
So  much  is  certain.  Now  let  us  suppose  that  he  chose  the  easy 
task  of  measuring  the  distance  between  the  summit  of  the  auro- 
ral arch  and  the  nearest  point  of  the  halo  instead  of  the  less 
simple  task  of  measuring  the  interval  separating  this  summit 
from  the  relatively  brilliant  moon,  in  which  case  he  would  have 
been  obliged  to  experiment  with  the  dark  glasses  which  are  not 
well  adapted  for  this  kind  of  work.  Under  this  supposition  and 
the  assumption  that  the  halo  was  the  ordinary  one  of  22°,  we 
find  that  the  distance  separating  the  apex  of  the  arch  and  the 
moon  was  105°. 2,  which  accords  well  enough  with  the  astro- 
nomical fact.  The  only  other  modification  necessary  is  to 
assume  that  the  circle  which  intersected  the  auroral  arch  a  few 
degrees  to  the  west  of  its  vertex  was  the  vertical  circle  through 
the  moon  instead  of  the  circle  which  accompanied  the  moon. 
With  these  highly  plausible  assumptions  the  records  of  a  trained 
observer  are  made  perfectly  clear  and  probable,  while  without 
them  they  are  entirely  self -contradictory ;  yet  with  these  modi- 
fications the  last  bit  of  confirmatory  evidence  for  the  ninety- 
degree  halo  of  Hevelius  falls  to  the  ground.  It  does  not  seem 
uriphilosophical  to  conclude  that  an  inexplicable  phenomenon 
recorded  only  once  in  a  quarter  of  a  millennium  does  not  really 
exist. 


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